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Full text of "Physics"

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Within the last twenty years the methods of teaching physics ' 
have been revolutionized. The reaction against the loose and 
desultory methods previously in vogue was started by the em- 
phasis given to laboratory work in the new books which then 
appeared. The movement gained impetus from the influence 
brought to bear on 'the schools by the Harvard Entrance Require- 
ments and the Report of the Committee of Ten. This pressure 
on the schools resulted in a demand for closer observation by the 
students, involving careful measurements. 

As far as science is concerned, the most important result of 
this introduction of laboratory work into the schools has been the 
development in the public mind of a widespread recognition of 
the fundamental principle that knowledge is real and living to the 
individual, only when it is founded on personally observed facts 
and personal experience. 

Nevertheless, during the past few years it has become clear 
that the present methods of applying this principle to instruction 
in physics have failed to arouse in the students that real enthusi- 
asm for the pursuit of the subject, which is indispensable for the 
mastery of its principles. This has been found true in spite of 
the fact that, in their own way, boys and giris have by nature and 
disposition the keenest interest in physical phenomena. 

In recognition of this fact, many attempts have been made 
to develop a better system of instruction. Such a system ought 
to give the mental training that has been so much emphasized; 
but it ought also to inspire in the boys and giris a living enthusiasm 
for the subject, and to develop in them the scientific habit of 
mind, the ability to utilize their knowledge, and a just apprecia- 
tion of the significance of natural phenomena. If physics can 
be so taught as to develop in the student these elements of 
power, so vital to his future career, there can be no doubt that 
in due time its educative value will • be properiy appreciated and 

its popularity restored. 



This book is the contribution of two of the fraternity of teachers 
toward the attainment of this end. The method of instruction 
herein set forth has been developed after long experience and 
much experimenting with high school and college classes. 
The greater part of the book has been in manuscript for more 
than two years, and has been used in connection with class 
teaching. The results obtained have been so encouraging that 
the book is submitted to other teachers, in the hope that it may 
be of service to them. 

In a recent work, President G. Stanley Hall comments at 
length on the decline of interest in physics in the high schools, 
and consequently also in the colleges, and he' suggests several 
remedies.^ He attributes this failure of physics to "the violence 
done to the nature and needs of the youthful soul by the present 
methods and matters." He points out that this violence consists 
in: 1. Neglect of the hero-ology of the science, of historical and 
biographical references, so that the learner is not made to "feel 
vividly a sense of growth." 2. "The rage to apply mathematics 
to the boy's brain processes," instead of appealing to his interest 
in concrete things. 3. The failure to realize that "very much 
thoroughness and perfection violates the laws of youthful nature 
and of groA\i;h." The young student "wants only answers that 
are vague, brief, but above all suggestive." 4. Neglect of the 
practical side of the subject, which is the side that appeals most 
strongly to the youth. **He is chiefly interested in the 'go' of 

The methods of instruction which have proved helpful to us, 
and which are embodied in this book, are in harmony with many of 
President HalFs suggestions. We have endeavored to strengthen 
the presentation of the subject, and aid the teacher in three ways: 
^ I. By arousing interest. II. By developing the scientific habit of 
! thought. HI. By presenting some of the principles from the 
historical standpoint. Some of the ideas that have guided us in 
this endeavor are the following: 

I. Interest. Interest is rarely stimulated in youth by elegant 

* Adolescence, by G. Stanley Hall, Vol. II, pp. 154 seq., New York, 
AppletOQ, 1905. 


and abstract mathematical treatment; nor is it often aroused 
by rigorous logical demonstrations. It is aroused by beginning 
with some concrete thing that goes — ^something which is already 
familiar. Interest may be sustained by basing the discussion on 
these familiar and concrete things. Nothing helps more than 
to have the student feel that you are discussing with him some- 
thing concerning which he already knows a little, and of which he ; 
has long been desirous of knowing more. The authors believe that 
the adoption of an informal style and the use of arguments that i 
are physical, rather than mathematical, will also be helpful; for 
they have been mindful of the success of the great teachers, Fara- 
day and Tyndall, in imparting scientific ideas to untrained minds 
in this way. Mathematics is an excellent servant but a very 
bad master; so equations are used only where they are clearly a 
help to the student, and the development of each is carefully 
presented with the aid of physical, rather than mathematical 

The aim has been to show the student that knowledge of physics : 
enables him to answer many of the questions over which he has ) 
puzzled long in vain. He is approached with the attitude: What 
do the forces of Nature do for us, and how do they do it? His 
self -activity is stimulated by this questioning attitude of text-book 
and teacher, and he is urged to investigate independently at home. 
His interest is not killed at the start by attempting to cram him 
with definitions of things to which he has no corresponding con- 
cept, such as indestructibility, impenetrability, and the like. 
Nor is he deceived by attempted definitions of undefinable con- 
cepts, such as mass, force, time, and space. On the contrary, 
the attempt is made to implant the concept and create the demand 
for its name, or definition, which is withheld Until the need for it ' 
is apparent. 

II. The Scientific Method. Although interest may be ob- 
tained through the technical applicati^s of physics, the teaching 
must not consist in descriptions of these only, any more than of 
descriptions of laboratory apparatus only. The attainment of 
scientific principles is always the purpose or end of the argument; 
not inventions, nor yet laboratory experiments. Science must 


be shown to consist in that body of organized knowledge which 
makes invention possible. Beginning arguments with inven- 
tions, or general observations of phenomena, may not be the 
logical order, but it is more nearly the order in which Nature 
herself teaches, and the result of the argument does not lose in 
definiteness, clearness, or accuracy^ provided the laboratory is 
continually held up as the final court of appeal where all doubtful 
questions are settled. 

Each chapter in this book is a continuous argument toward 
some principle or principles, and the entire book is an argument 
toward the conclusions stated in the last chapter. This treatment 
; is intended to develop and foster the habit of scientific thinking. 
The attempt is made (1), to interest the student in observing care- 
fully and accurately first the familiar things about him, and then 
the things in the laboratory; (2), to interest him in detecting analo- 
gies and similarities among the things observed; (3), to train him in 
I keeping his mind free from bias and in drawing conclusions 
( tentatively; (4), to make him see the value of verifying the con- 
i elusions and accepting the result, whether it confirms or denies. his 
inferences. The arguments in the various parts of the book are 
not all alike; there are many forms in which the scientific method 
may be used. 

We have tried deliberately to give the student the impression 
that science leads to no absolute results — that, at best, it is merely 
a question of close approximation; of doing the best we can, and 
accepting the result tentatively, until we can do better. This 
attitude places the teacher also in the position of a learner and 
prohibits him from making use of didactic or dogmatic statements; 
for these are the bane of science as well as of other things. Science 
instruction, that does not develop mental integrity, freedom of 
the personal judgment, and tolerance, fails in a very vital spot. 

III. History. References are given to books in which the 
biographies of the great men of science may be read, and the 
student is urged to read them and report. The arguments used 
by some of the great thinkers have been briefly sketched, and 
the methods devised by them for reaching conclusions have been 
given. The attempt has been made to present them as they live 


in the ideas which they have handed down to us; to picture their 
mental processes and attitude, and to show how one thing leads 
to another as the subject develops in the discoverer's mind. 

We wish to call the attention of our colleagues to several prac- 
tical points. In the first place, although each chapter is a con- 
tinuous argument, the paragraphs are headed in black type, 
so that the important steps are well marked; and a summary and |^ 
set of questions are added at the end of each chapter, to assist T 
the student in fixing the subject-matter in mind. The teacher 
will, we think, find these latter very helpful to his pupils in both 
advance and review work. 

In the second place, the continuity of the treatment is not 
interrupted by the insertion of descriptions of laboratory and 
lecture experiments in fine type. Judged from our own experience, 
such experiments, thus inserted, confuse rather than assist the 
student. It goes without saying, that we expect both laboratory ( 
and lecture experiments to be given in connection with tliis book;y 
but every laboratory experiment made by the student, and* every 
experimental demonstration by the teacher should have a definite \ 
relation in time, place, and subject matter to the general argument \ 
as presented in the text. An experiment is simply an incum- 
brance and a source of distraction to the student unless its rela- { 
tion to the general scheme of the lessons in the classroom is per- ' 
fectly obvious. A detailed description of a lecture experiment 
which he has not seen is of relatively small value to the student, 
and ordinarily there is no interest or profit to him in obtruding 
on his attention the distracting details of setting up and manipu- 
lating the apparatus. If such description of an experiment 
occurs in the text book, while the teacher chooses to make it 
with some other style of apparatus, different in its details, his con- / 
fusion is all the worse, for his attention is distracted from the( 
principle to be illustrated, and lost in the details of the apparatus. 

On the other hand, when the student is to make an experi- 
ment himself in the laboratory, he must be given many details in 
order that he may manipulate, observe, and record successfully 
and without loss of time. It is the province of the laboratory \ 
manual to give these details, for they can not be included in a text ) 


book without encumbering it to the exclusion of important theo- 
retical matter, and destroying its unity. We have therefore pre- 
ferred to leave the choice of illustrative experiments largely in the 
hands of the teacher, who may thus select them according to his 
individuality, his equipment, and the circumstances and limita- 
tions of his class and community. 

I We have bnsed the argument wherever possible on the pupils' 
1 experience, expecting this to be supplemented by the teacher 
with lecture demonstrations and laboratory experiments, chosen in 
accordance with the conditions which he has to meet and with 
his own taste and judgment. But when a particular kind of ex- 
perimental evidence is necessary to the argument, it has been used, 
without manipulatory details and in uniform type with the other 
subject matter. 

In the third place, many of the old and familiar landmarks of 
the elementary physics text do not appear in these pages. Among 

f these may be mentioned the division of levers into classes; the 
wedge; the classification of equilibrium as stable, unstable, and 
neutral; specific gravity as distinguished from density; the elec- 
trophorous and the electrostatic machine; the concave and con- 
vex mirrors; multiple reflection; and the formulas concerned 
with the radii of curvature of lenses. These have been omitted 
because they seem of less interest and importance than the' 
following new subjects which we have been able to introduce in 
the space thus saved: The use of graphical methods and of 
vectors; the discussion of efficiencies of engines, both prac 
tical and theoretical; the relations among electrostatic charge, 
current, and magnetic field; the meaning of harmony; the nature 
of spectra; the reasons for the electromagnetic theory of light; and 
the electron theory of matter. We also believe that the presenta- 
tion of the subjects of rotary motion and of optical instruments 
will be found much simpler and more satisfactory than those 
usually given. 
/ The problems are also an innovation. They include no 
J cases of forces a, 6, and c, meeting at a point q, etc., but are, 
. as far as possible, real, concrete cases, such as occur in actual 
practice, and which every boy or girl ought to know how to meet. 


( They also contain many of the subjects usually placed in the text 
I and there explained; for example, the pulleys, distillation, and the 
Wheatstone bridge. We hope that this form of problem will 
Unterest the student, as most of them are problems in whose solu- 
tion he can see some use. 

Other devices for catching and holding the interest are the 
questions and the suggestions to students at the end of each chap- 
ter. We hope that these latter will be stimulating to the students 
and serve as hints which will lead them to suggest for them- 
selves other home experiments. Are not such experiments, clumsy 
though they be, yet made with a genuine interest in finding 
out something — in getting the answer from Nature herself — far 
more useful than many that are made in some laboratories? 

The illustrations are also a novelty. Great pains have been 

'^(taken to have every picture a photograph of a real thing, for a 

photograph is always more interesting than a woodcut. It is 

believed that these will add much to the interest of the work. 

We have been favored with the original photographs for 
many of these illustrations, by the firms and individuals men- 
tioned on page x, whom we wish to thank for their courtesy. 

We also desire to express our thanks to Professor R. D. 
Salisbury of the University of Chicago, Editor-in-Chief of the 
Lake Science Series for many valuable suggestions, and to 
Messrs. A. A. Knowlton of the Armour Institute of Technology, 
J. H. Kimmons of the Austin, Chicago High School, and 
C. Kirkpatrick of the High School, Seattle, Washington, for aid 
in the reading of the proof. 

Many of the line diagrams are new and have been designed 
and executed with much thought and care, so as to present the 
essential ideas without complication by unnecessary details. 

That great difficulties are involved in the working out of a 
method of instruction differing in principle from that in general 
use must be apparent to every one. We know better than any one 
else can that we have not produced a perfect book. This might 
be approximated by the concerted action of all teachers of physics. 
We therefore hope that members of the teaching fraternity will 
regard the result of our work as a first approximation, and will 


join with us in making a united effort to lift our subject up to 
its proper place, and to inspire our young friends with an adequate 
appreciation of its interest, its majesty, and its grandeur. To 
this end we appeal to our colleagues to give us the benefit of their 
experience by sending us suggestions and criticisms, which will be 
gratefully received and carefully considered. 

Charles Rtborg Mann, 
George Ransom Twiss. 


^ Plate I. The Lake Shore and Michigan Southern Railway. Large 

copies of this picture in color may be obtained for 50 cents, by 

applying to Mr. A. J. Smith, General Passenger Agent, Cleveland, 

Fig. 11. The Electric Vehicle Co., Hartford, Conn. 
Figs. 16, 17, 18. The Eastman Kodak Co., Rochester, N. Y. 
Fig. 19. Pawling, Harnischfeger & Co., Milwaukee, Wis. 
Plate II, and Figs. 70, 156, 157, 158. The Niles-Bement-Pond Co., 

New York. 
Fig. 31. The Manitou and Pike's Peak Railroad Co,, Manitou, Col. 
Fig. 51. Crowe Bros., House Movers, Chicago, 111. 
Plates III, IV, VI. The AUis-Chalmers Co., Milwaukee, Wis. 
Fig. 61. The Bausch and Lomb Optical Co., Rochester, N. Y. 
Figs. 65, 72. The Ingersoll-Sargeant Drill Co.. New York. 
Fig. 74. The Chicago Bridge and Iron Works Co., Chicago, 111. 
Fig. 77. The Century Co., New York. 
Figs. 91, 92. The Whitlock Coil Pipe Co., Hartford, Conn. 
Fig. 100. The Otto Gas Engme Co., Philadelphia, Pa. 
Fig. 101. Mr. Alfred Stieglitz, New York. 
Plate VII, and Figs. 102, 131, 132, 133, 141, 142, 143, 144, 147, 

148, 151. The Westinghouse Electric Co., Pitlsburg, Pa. 
"The Electric Spark in Nature," page 205. Mr. M. I' Anson, 

Newark, N. J. 
Fig. 162. The Electric Controller and Supply Co., Cleveland, Ohio. 
Figs. 168, 169, 170. ' The Electric Storage Battery Co., Philadelphia, 

Plate VIII. The University of Chicago, Chicago, 111. 
Figs. 236, 237. Wm. Scheidel & Co., Chicago, lU. 



Introduction 11-14 

Motion, Velocity, Acceleration — 

Motion of a train — How velocity is measured — Units — 
Graphical representation of velocity — Analytical repre- 
sentation of velocity-^Slope — Changing velocity — Accel- 
eration — Graphical and analytical representation of accel- 
eration — Measurement of acceleration — Summary — Ques- 
tions — Problems — Suggestions to students 15-32 

Mass and Energy — 

Production of acceleration —Acceleration and force — Dif- 
ferent bodies having the same acceleration — Mass — 
Masses compared by forces — Relation of force, mass, and 
acceleration — Unit mass — Weight — Galileo*? experiment 
— Weight and mass — Density — Work, force and distance — 
Unit work — Energy, how measured — Efficiency — Kinetic 
and potential energy — Newton's laws of motion — 
Power — Engineering units — Summary — Questions — 
Problems — Suggestions to students 33-56 

Composition and Resolution of Motion — 

Up grade — Composition of motions — Vectors — Motions at 
right angles — Motions not at right angles — Vector solu- 
tions — Analytical solution — Traveling crane — Resolution 
of motions — Force vectors — Balanced forces — Mechanical 
advantage — Summary — Questions — Suggestions to stu- 
dents 57-72 





Moments — 

How rotation is produced —Moment of force — The lever — 
Work done by the lever — The lever principle — Parallel 
forces — Weight and center of mass — Equilibrium — Sta- 
bility, how measured — Determination of the center of 
mass — Mechanical advantage of a composite machine — 
The law of machines — The screw — The equal arm bal- 
ance — Review — Summary — Questions — Problems — Sug- 
gestions to students 73-96 

Rotation — 

Flywheels — Angular measurement and units — Correspond- 
ence with linear measurements and units — Moment of in- 
ertia and mass — Determination of moment of inertia — 
Conditions for circular motion, centripetal force— Burst- 
ing wheels — Distribution of mass — Moment of mass — 
Railroad curves — Spinning tops — Summary — Questions — 
Problems — Suggestions to students 97-111 

Fluids — 

Pumps — Air has weight — Torricelli's experiment — Pascal's 
experiment — Mercurial barometer — Characteristics of 
fluids — Pascal's principle — Hydraulic machines — Free 
level surface of a liquid — Gases — Air pump— Guericke 
— Density of air — Theory of pumps — Archimedes' prin- 
ciple — Flotation — Buoyancy — Determination of density 
— Boyle and his law — Summary — Questions — Prob- 
lems — Suggestions to students . 112-136 

Heat — 

Heat and work — Thermometers — Temperature scale — 
Gases — Change of volume at constant pressure — Change 
of pressure at constant volume — Air thermometer — Ab- 
solute temperature — Expansion of solids and liquids — 
Heat quantity — Gram calorie — Specific heat — Steam — 
Evaporation — Pressure and temperature of saturated 
vapor — Boiling point — Superheated vapor — Critical tem- 
perature — Humidity — Dew — Latent heat — Water and 
climate — Summary — Questions — Problems — Suggestions 
to students 137-157 




Transfer op Heat — 

Conduction and convection — Applications — Radiation — 

Diffusion — Evaporation — Gaseous pressure — Effect of 

heating — Kinetic hypothesis — Radiation — The ether — 

Prevost's theory of exchanges — Absorption— Absorbing 

power of water vapor — Radiation and absorption — Heat 

and light — Summary^-Questions — Problems — Suggestions 

to students 158-170 

Heat and Work — 

Mechanical equivalent of heat — Gas is heated when com- 
pressed — Gas cools when it expands and does work — 
Liquid air — Cooling by evaporation — Manufactured ice — 
The steam enghie — Work done by the steam — The pres- 
sure-volume graph — Back pressure — Lower pressure at 
exhaust — Condensers — Higher boiler pressure — Heat 
energy consumed — Efficiency and temperature — Com- 
parison of efficiencies — The triple expansion engine — 
The gas engine — The steam turbine — Summary — Ques- 
tions — Problems — Suggestions to students 171-189 

Electricity — 

Transmission of power — Generators — Early knowledge of 
electricity — Gilbert — Electrification — Conductors and in- 
sulators — Repulsion — Discharge — Electroscope — Both . 
bodies equally charged — Two kinds of charge — Polariza- . 
tion — Charging by influence — Charge on the outside of a 
conductor — Coulomb's law — Leyden jar — Condensers — . 
Operation of a condenser — Discharge of condenser is oscil- 
latory — Lightning — Summary — Questions — Problems — 
Suggestions to students 190-210 

Magnetism — 

Lodestone and compass — Magnetic curves — Magnetic 
field — Like poles — Unlike poles — Permeability — Magnetic 
circuit — Earth's magnetism — Unit pole — Law of magnetic 
force — Chief characteristics of magnets — Electric currents 
— Voltaic cell — Electromagnetism — Magnetic field of the 
current — Electromagnets — ^Telegraph — Relay — Grounded 



wires — Electric bell — Galvanometers — A suggestive ex- 
periment — Motors — Motor parts — From toy to practical 
machine^ Ampere's theory of magnetism — Magnetic field 
of moving charges — Energy of a magnetic system — Sum- 
mary — Questions — Problems — Suggestions to students. . . 211-243 

Induced Currents — 

Sources of current — Current and magnetic field — Faraday's 
discovery — Current induced by a moving magnet — Num- 
ber of lines of force changed — Currents induced by cur- 
rents — Iron core — Laws of induced currents — ^The dynamo 
principle — The dynamo — Magnetos — Alternating current 
dynamos — The induction coil — The transformer — Alter- 
nating current motors — The telephone — Summary — 
Questions — Problems — Suggestions to students 244-264 

The Electric Current at Work — 

Pressure and current in the arc lamp — Current strength — 
Resistance — Laws of resistance — Ohm's law — Ammeters 
and voltmeters — Electric power — Watt meters — Arc light 
plant — Incandescent lamps, parallel distribution — Incan- 
descent light plant — Heating effects of the current — Joule's 
law — Heat loss in transmission — Three wire system — Al- 
ternating current transmission — Divided circuits — Shunts 
— Arc lamp regulation — Lifting magnets — Voltaic cells — 
Energy of the cell — Polarization of cells — The ion hypoth- 
esis—Commercial cells — Electrolysis — Faraday's laws — 
Electroplating — Storage batteries — ^^ Retrospect — Sum- 
mary — Questions — Problems — Suggestions to students . . . 265-299 

Wave Motion — 

Water waves — Origin of waves — Characteristics of waves — 
What waves tell us — Wave motion — Wave length — 
Period — Phase — Velocity of propagation — Waves of sim- 
ple shape — Complex waves — Waves of different shapes — 
Stationary waves — Summary — Questions — Problems — 
Suggestions to students 300-316 




Simple Harmonic Motion — 

Uniform circular motion — Displacement and force — The 
sine curve — Period, mass, and force constant — Pendu- 
lum — Uses of the pendulum — The Foucault pendulum- 
Summary — Questions — Problems — Suggestions to stu- 
dents 317-327 


Sound — 

Sources of soimd — Soimd a wave motion — Soimd waves 
longitudinal — Velocity of sound — Resonance — Noise — 
The piano — Pitch — Musical intervals — Laws of strings — 
Vibrating rods — Tuning forks — Organ pipes-^Air columns 
as resonators — Intensity — Summary — Questions — Prob- 
lems — Suggestions to students 328-341 

The Musical Scale — 

Development of the musical scale — The related triads — 
The vibration numbers — The major scale — The complete 
scale — The tempered scale — Standard pitch — Forced vi- 
brations — The ear — Beats — Discord due to beats — Sum- 
mary — Questions — Problems — Suggestions to students . . . 342-356 

Harmony and Discord — 

Wave shape and tone quality — The vibrating flame — Mus- 
ical tones complex — How musical tones are possible — Fun- 
damental and overtones — Overtones of strings — Reson- 
ators — How the ear perceives a complex tone — Related 
tones — Chimes — Summary — Questions — Problems — Sug- 
gestions to students 357-368 

Light — 

What does light do for us — Direction — Image by a pin . 
hole — Image by a lens — The eye — How light is changed in 
direction — Refraction — Index — How the lens forms the 
image — Reflection — Diffuse reflection — Summary — Ques- 
tions — Problems — Suggestions to students 369-383 




Optical Instruments — 

Principal focus — Image of a point source — Construction of 
the image — Lens angle — Size and distance of the image — 
Virtual image — How the eye is focused — Spectacles — The 
simple microscope — The camera — Stops — Spherical aber- 
ration — The telescope — The concave lens — The opera 
glass — The compound microscope — Magnification — Reso- 
lution — Smnmary — Questions — Problems — Suggestions to 
students ' 384-402 


Color — 

Newton's experiment — Interference fringes — Wave length 
and color — Interference in white light — Dispersion — The 
spectrum — Bright-line spectra — Measurement of disper- 
sion — Achromatic lens — Spe.ctrum analysis — How the eye 
perceives color — Mixing colors — Colors of ordinary ob- 
jects — Paints and dyes-^Mixing pigments — Summary — 
Questions — Problems — Suggestions to students 403-422 

Velocity op Light — 

. What we can learn from the velocity of light — Galileo's 
method — Fizeau's method — The velocity determined — 
Velocity of electric waves — Wireless telegraphy — Light 
and electricity operate through the same medium — The 
complete spectrum — Summary — Questions — Problems — 
Suggestions to students 423-434 

Electrons — 

Light waves start at an electrically charged particle — Other 
such particles — Cathode rays — Action of magnet — deter- 
mination of ~ — Comparison with the ion of electroly- 
sis — The electron — Radioactivity — The X-rays — The na- 
ture of white light — Conclusion 435-446 

Index 447-453 



It has been said that man made his sta,rt on the long road 
toward enlightenment when he learned how to make a fire. For 
many centuries, our ancestors groped at a snail's pace along this 
road where we of the twentieth century are advancing by leaps 
and bounds. 

By slow and painful steps, prehistoric man learned to use fire 
in order to keep himself warm, to cook his food, to get metals out 
of their ores, and to forge them into rude tools and weapons of de- 
fense. By means of signal fires on the hilltops, he sent his first 
wireless messages across the valleys. The magnetic force of the 
lodestone and the electric attraction of amber were known to the 
ancients, and the fact that steam pressure can be made to produce 
motion was known in the early centuries of our era. Why was it 
that so many centuries elapsed before man learned to subdue these 
forces of nature and make them do his will? Now we have the 
steam engine, the electric dynamo and motor, the power printing 
press, the power loom, the telephone, the wireless telegraph. By 
means of these and countless other inventions, one man can do the 
work of hundreds, the continents are linked together, darkness is 
turned into light, time and space are vanquished. 

We can best realize how important are these inventions when we 
try to think how we should get on without them. And yet this great 
development of miracle working machinery has come within the 
space of three centuries, and the greater part of it within seventy- 
five years 1 

The stories of how these marvelous inventions came to be, of the 
struggles of the men who brought them into being, and of the pa- 


tient researches and brilliant discoveries of the men of science who 
established the foundation principles upon which all these inven- 
tions rest are among the most important and most interesting chap- 
ters of history. 

In the studies which follow, we shall endeavor to get an 
understanding of some of these principles, to gain at least a slight 
acquaintance with some of the great discoverers who formulated 
them, and to get some insight into the kind of thinking and the 
methods of experimentation by which their truth has been made 
plain. Such studies are of interest not only to those who expect 
to make practical use of them, but also to those who, in the pursuit 
of a liberal education, wish to learn how to think clearly, to ex- 
press themselves precisely, and to test their conclusions accurately, 
as well as to get a properly balanced view of human life and ac- 

The principles of physics are most easily understood by the 
beginner, and are also most interesting, when they are studied in 
connection with his own experiences. For no one can live long in 
this scientific age, surrounded as he is on all sides by the fruits of 
discovery and invention, without having a large amount of experi- 
ence with the forces of Nature and without obtaining therefrom a 
large fund of general information. 

A rapidly-moving railway train is certainly a familiar object to 
every one. Even a small child would not have to be told that 
Plate I is the picture of such a train. Moreover, we all know that 
the locomotive causes the train to move, and that it can not do so 
unless it has a fire in it. We are also familiar with the fact that 
the locomotive must be supplied with water, and that in the boiler 
this water is converted into steam, which somehow makes the big 
driving wheels turn. That such an engine warns us of its approach 
by means of a whistle and a bell, and that it lights its own path 
in front of it at night by means of a brilliant headlight, are well- 
known facts. 

Now, although these and many other things about the locomo- 
tive are matters of everyday knowledge to most of us, how many 
of us can tell exactly how the steam makes the engine's driving 
wheels turn? And why is steam used at all? Why are some loco- 


motives large while, others are small? How does the whistle 
work, and how does its sound get to us? How is the headlight 
made to send its light forw^ard on the tracks? 

A ride in a steam or trolley car is one of the most common of 
our experiences, and we all know that the car has many different 
kinds of motion. Wherein do these motions differ? How are 
speeds measured and compared with one another? What sort of 
velocity has the car while it is starting or stopping? Why are 
we thrown against the side of the car when it rounds a curve? 
How is it that some engines can go faster than others? 

The picture of the Twentieth Century Limited (Plate I) was 
taken while the train was running at full speed. How do cameras 
and lenses work? 

We can obtain the answers to these and to other similar questions 
without great difficulty, if we are willing to devote to the subject 
some careful study and thought. When we have done this, w^e 
shall find that the knowledge thus acquireti gives us a greater con- 
trol over the forces of Nature, and that the training thus obtained 
is of great service to us in everything we may wish to do. 



• • 

1. The Motion of a Train. In order to find the answe^^fci 

some of the questions just asked, let us suppose that a locomdtk'f " 
stands with steam up, ready to make the run to the next station/ 
When it starts, we notice that at first it moves slowly, and that its 
velocity gradually increases until it has attained **full speed," when 
it runs for some time at a rate that is nearly constant. As the 
next station is approached, the speed gradually decreases; and the 
train comes to a full stop. What can we leam of its motion, of the 
way in which it rounds curves, of how it gets up speed, and of 
how it stops? How shall we describe and measure its velocity, 
and how take accbujit of the energy that it must expend in order 
to move its load? 

2. How Velocity is Measnred. Since all motion implies 
both distance and time, and since distances and times must be meas- 
ured in order to be compared, it is necessary to have units of length 
and of time in terms of which the measurements can be expressed. 

The units adopted in all scientific work are 
purely arbitrary, and are chosen simply for con- £~\ r~~]^, 

venience. The unit of length is the centimeter, \ \ / / g 
which is the one-hundredth part of the dis- \ V — / / J 
tance between two lines on a certain bar of plati- / / V \ ^ 
num-iridium when the bar is at the temperature p / \ n j 

of zero degrees Centigrade. This bar is care- fig. i 

fully preserved at Paris, and is called the Inter- Standard Meter^ 
national Prototype Meter. The symbol for 
centimeter is cm, and that for meter is m. The unit of time is 
the SECOND, which is the one-eighty-six-thousand-four-hundredth 
of the mean solar day. Its svmbol is sec. 

Now if a train, moving uniformly, attains in one minute a 


16 y *. tPHYSICS 

distance of 150,00g cpi*from a given post in a certain direction, 
then in one sepgft>3L*tfce change in its distance in that direction 
from the post*. wiH *be -^^ of 150,000 or 2500 cm. Therefore it 
travels at gti^e rate that its distance from the post changes 2500 
cm eveiy'spftond. This rate of change of distance is called linear 


^ .. K'^without changing the direction of its motion, a body trav- 
. if^ equal distances in equal times, no matter how small the time 
• .in'tervals are taken, its velocity is uniform or constant. The unit 
•^•'of velocity used in physics is the velocity of a body moving uni- 
formly over one centimeter each second. If the body traverses 
2 cm in each second, its velocity would be two units, or 2 centi- 
meters per second, and so on; therefore, a unijoria velocity is 
measured by the number of centimeters passed over in one second, 

3. Comparison of Velocities. Let us now compare the veloc- 
ity of our train with that of a fast freight which passes over a 
distance of 90,000 cm from a given post in onfe minute. Its veloc- 
ity is then ^V of 90,000 = 1500 cm per second, which is evidently 
I of 2500, the velocity of the express. Similarly, a rifle ball that 
passes over 240,000 cm in 3 seconds, has a velocity of 80,000 cm 
per second. In all of these cases we obtain the number that 
expresses the velocity by dividing the number of centimeters in 
the distance by the number of seconds in which that distance is 
traversed. Since the expression for velocity is thus obtained, an 
appropriate sjrmbol for linear velocity is ~. Symbols made in 
this way will be found very useful because they show at once how 
a quantity like velocity is expressed in terms of the fundamental 

. units. 

4. The Analytical Method. Uniform velocity may therefore 
be measured by dividing the distance passed over in a given 
time by the number of seconds in that time. Since this expression 
is rather cumbersome, it is more convenient to state it in an 
abbreviated form by means of algebraic symbols. This is done 
by letting v represent the linear velocity, I the number of cm in 
the distance traversed, and t the number of sec in which the 



distance is traversed. We may then write the expression 

... Distance passed over in a ffiven time . . „ 

uniform velocity = — .r^ — -, — ^—^ i — : — " , .. m the form 

•^ Number oi seconds m that time 


This is the equation for uniform motion. This method of express- 
ing relations by means of an algebraic equation is called the 
ANALYTICAL METHOD. This method is extensively used in physics 
and engineering, and has the advantage of great conciseness. 






- - 







- - 

■ — 
















" - 











4 6 8 
Time In Hours 

4 6 8 10 

Fig. 2. Variations of Temperature 
During One Day 

5. The Graphical Method. There is another very convenient 
method by which relations of this kind are expressed. This 
method is familiar to every- 
body, since it is very gener- 
ally employed to picture the 
relative variations of two 
quantities, both of which 
are continuously changing 
in value. Thus Fig. 2 repre- 
sents the variations of tem- 
jjerature during a day. The 
time intervals are repre- 
sented by horizontal dis- 
tances, and the corresponding temperatures by vertical distances. 
A single glance at the diagram tells us whether the range of 
temperature on that day was large, when it was highest, when 
lowest, and how hot or cold the air was. This method of present- 
ing relations is called the graphical method. 

Let us then apply this method to the train mentioned in Art. 2. 
Since in this case the two quantities that vary are time and the 
distance of the train from the given post, we must let one of the 
quantities be represented by horizontal distances, and the other 
by vertical distances. We may choose freely what scale to use, 
i.e., how great a length on the diagram shall represent a sec or a 
cm. In this case we shall get a drawing of convenient size if we 
let 1 cm in the horizontal direction OX represent 1 sec, Fig. 3; and 
1 cm in the vertical direction OY represent 2000 cm. 



If now we begin to consider the motion at the instant when 
the front of the engine, going at the rate of 2500 ~, passes a cer- 
tain post: then at that instant, since no time has elapsed and no 
distance been passed over, i.e., the time is zero and the distance 
also zero, the corresponding point on the diagram will lie at 0, 
Fig. 3. At the end of one second the train is 2500 cm from the 
post. Therefore the point that corresponds to this condition 

must represent a time of 1 sec and 
a distance of 2500 cm, and so 
must be 1 cm from OY in the 
direction OX, and 1.25 cm from 
OX in the direction OY, To locate 
this point we lay off 1 cm along 
OX to a?!, and draw from x^ a 
dotted line parallel to OY. We 
then lay off 1.25 cm along OY to y^ 
and draw through y^ a dotted line 
V parallel to OX, The intersection p^ 
— ^of these two dotted lines will then 
be the point sought, since it is 1 cm 
from OY and 1.25 cm from OX. 
In like manner, at the end of the second second the train is 
5000 cm from the post. So we lay off 2 cm along OX to the point 
X2 to represent 2 sec, and 2.5 cm along OF to the point ^2* *o 
represent 5000 cm. We then draw the dotted lines as shown in 
the figure, and find the point p2, which therefore represents the 
conditions at the end of 2 sec. Similarly, the point pg, distant 
3 cm from OY and 3.75 cm from OX^ represents the conditions 
at the end of the third second; and so on. Note carefully that 
the line obtained does not represent the path of the train. 

We next draw the straight lines Op^, p^ p^, pa Vzj ^tc. Is the 
resulting line Opg straight? Do the points that represent the con- 
dition of the train's motion at 0.5, 1.7, 2.2, 2.5 sec also lie on this 
line? Is there on the line a point corresponding to every possible 
instant of time? Does every such point also represent a distance 
from the post? Does the time Opg completely represent the 
motion of the train with respect to both distance and time? 

Fig. 3. 

1 \2 3 

Uniform Velocity 



Since we shall often use the graphical method, we shall need 
to know the names of the lines and points. The two lines OX 
and OF are called coordinate axes. The distances Ox^, Ox^y 
Ox^j etc., are called abscissas. They may be measured from any 
point on OF along a line parallel to OX; thus y^ pi, = Ox^, 
2/2 P2i = 0^2> ^t^- The distances Oy^, Oy^y Oy^, etc., are called 
ORDi nates, and may be measured from any point on OX along a 
line parallel to OF. OX is called the axis of abscissas, and OY 
the AXIS OF ORDINATES. is Called the origin of coordinates. The 
line O/jg representing the relations considered is called a graph. 

Fig. 4. 
Slope Indicates Velocity 

6. What the Slope Indicates. Let us now add to our diagram 
graphs for two other trains, one of which is a fast freight F 
traveling uniformly at the rate of 
1500 ^, and the other an express E at the 
very high speed of 3000^. The result 
is sbo\\Ti in Fig. 4. In what respect are 
the second and third graphs like A, the 
first? Which graph has the steepest 
SLOPE, or in other words which makes 
the greatest angle with the axis of 
abscissas? Would the graph for a slow 
freight traveling at a rate less than 
1500^ have a greater or a less slope 
than that of the fast freight? Would the graph RB for a rifle ball 
having a speed of 80,000 ^ make a greater or smaller angle with 
the axis of abscissas than does that for the express? What charac- 
teristic of the motion is indicated by the steepness of the slope? 

It thus appears that in the graphical method of representation 
the velocity is represented by the slope, while in the analytical 

method (c/. Art. 4) it is measured by the ratio — . The slope and 

the ratio — then serve the same purpose. But on the graph / is 

represented by the vertical distance pn (Fig. 5), and t by the 
horizontal distance On. Therefore the slope of the line Op may 

be appropriately measured by the ratio ^. But pn is the side 



opposite the angle of slope, and On is the side adjacent to it 
in the right triangle pOn; and in a right triangle this ratio of 
the side opposite the angle to the side adja- 
cent to it is called the tangent of the 
ANGLE. It is clear that the tangent of a 
given angle has always the same value 
no matter what the size of the triangle is. 
Thus, since the triangles pOn and pqm 

. ., pn pm 

are similar, f—-=^—. 

On qm 

Hence the appro- 

priate measure of slope is the tangent of the 
angle that the graph makes with the axis 
^^°- ^' of abscissas. 

Measurement of Slope 

7. Increasing Velocity. Thus far we have considered the 
motion of the train only when it is uniform. What now are the 
characteristics of the motion 
just after the engineer has 
opened the throttle, so that 
the train is getting up speed; 
and what of the motion when 
he has shut off the steam and 
applied the brakes, so that 
the train is slowing down? 

Since, now, the velocity is 
changing at every instant, it 
can not be measured by the 
distance traversed in one 
second. Therefore the velocity at any instant is measured by 
the distance which would be traversed in one second, provided that 
throughout that second the rate were to continue the same as it 
was at the given instant. Suppose now that the train starts from 
rest, and that at the end of the first second it has gained a veloc- 
ity of 50 ^, that at the end of the second second its velocity is 
100 g^, and at the end of the third second 150 ^^; i.e., suppose 
that the velocities at the end of successive seconds are as fol- 

Fig. 6. Ready to Start 























Is the change of velocity for any one second the same as for any 
other second, i.e., is the change of velocity constant? If during 
the interval between the end of the eighth second and the end of 
the twelfth the velocity changed uniformly from 400 to 600 ^, 
what was the rate of change of velocity, i.e., the change of velocity for 
any one second? 

8. Decreasing Velocity. Again, let us suppose that when 
the train is slowing down, its velocity changes in the first 
second from 2500 to 2400 ^, and that at the ends of the successive 
seconds the velocities are as follows: 


cm ' 























What is now the rate of change of velocity? Since this rate of 
change is the ratio of the change of velocity to the time, it is 
expressed as a number of ^^ per second. Thus if the rate of 
change of velocity is such that 75 ~ is gained or lost each second, 
then this rate of change is expressed as 75 centimeters per second 
per second. It is customary to write this 75 ^^2- 

9. The Name Given to Eate of Change of Velocity is Ac- 
celeration. When the velocity is increasing, the acceleration is 
positive; and when the velocity is decreasing the acceleration is 
negative. When the acceleration is constant, as in the examples 
just givQn, the motion is called uniformly accelerated motion. 


It is to be noted that the expression for linear acceleration is 
obtained by dividing a number of units of velocity by a number of 
units of time, ' Since velocity is length divided by time, it is plain 
that acceleration is length divided by the square of time. Hence 
the symbol for the unit of acceleration is ^j- 

10. The Analytical Expression for Acceleration. The ana- 
lytical expression for acceleration may be found as in Art. 8, except 
that we now represent the related quantities by letters instead of 
by numbers. Thus, if a represent the acceleration, V the velocity 
at the end of a number of seconds denoted by t, and Vq the velocity 

V — V 
at the beginning of this time, then the acceleration is a = — - — ^. 

This equation is simply the definition of acceleration written in 
algebraic shorthand. 

It is often necessary to find the change in velocity in terms of 
the acceleration and the time. In order to do this, we multiply 
both members of our equation by <, thus obtaining the result 
V — Vq = at, i.e., the change of velocity is equal to a, the rate of 
that change, multiplied by t, the time. 

If we wish to find the value of the final velocity V when the 
other quantities are known, we add Vq to both members of this 
equation, which gives us 

V = Vf^-{-at (2) 

i.e., the final velocity is equal to the initial velocity 'plus the cJiange 
in velocity. 

11. Relation of Distances to Times. It will be interesting to 
know what sort of lines we shall get if we plot graphs that repre- 
sent the relations of distances to times while our train is starting 
and stopping. In order to do this we must first know the distance 
of the train from a given point at the end of each second. 

At the beginning of the first second, since the train is at rest, the 
velocity is zero: and the final velocity is this initial velocity plus 
the change, or F = i^o + ^^> ^s stated in equation 2, Art. 10. Now 
a, the acceleration, is 50 ^2 J hence the final velocity for the time 1 
sec isF = + 50X 1 = 50 ^. Since the velocity begins atO and 


ends at 50 ^, it must, during that first second, have all values 
from zero up to 50 ^. Which of these values may we use in calcu- 
lating the distance traversed in that second? Since according to 
our supposition the velocity increases imiformly, the train will 
traverse in a given time with the uniformly accelerated motion the 
same distance that it would have traversed during that time with uni- 
form motion at the average speed. The average or mean velocity, 
then, is that which we must choose. 

Since the velocity changes at a uniform rate, the average velocity 
may easily be found by taking the arithmetical mean of the initial 
and final velocities; and therefore, for the first second, if we represent 

this mean velocity by v, we have v = — ^ — ^ ^^ s^- 

Solving equation (1) for /, we have / = vt; and substituting, we 
get / = 25 X 1 = 25 cm, the distance of the train from the starting 
point at the end of the first second. Similarly for the time two 
seconds (since for each time period we must consider the motion 
from the beginning y in order to get the average velocity), the initial 
velocity is 0; and the final velocity is F= + 50 X 2 = 100 ^-^. 

Whence the average velocity 1;= = 50£I?; and the whole 

distance traversed up to the end of the second second is again the 
mean velocity multiplied by the time, i.e., I = vt =50X2 = 100 cm. 
In like manner, for 3 sec, we get F = + 50 X 3 = 150 ~, 

4- 1 ^0 

and V = \ = 75^, therefore / = 0;^ = 75 X 3 =f 225 cm. 

^ sec 

By the same method of calculation, we find that the distances 
for the first eight seconds are as follows : 










' 900 













12. The Graph for Distance and Time. We now have the 
data that we need, and can proceed to construct our graph. Let 



Fig. 7. Graph for Single Seconds 

US choose our scales so that for the abscissas 1 cm represents 1 sec, 
and for the ordinates 1 cm represents 100 cm. We locate the point 
corresponding to each second (Fig. 7) and find them to be for 
the beginning of the first second, pi for the end of the first second, 

P2 for the end of the second 

.„ . ^ second, p^ for the end of the 

third second, and so on. If, as 
before, we should connect the 
points in succession by straight 
lines, would the resulting line 
be straight? Does the velocity 
of our train change abruptly at 
the end of each second or is it 
increasing uniformly at every 
instant? Does the broken line 
connecting O, Pi,p2f etc., change 
its slope at every point or only 
at the points that we located? 
Then does such a line properly represent the uniformly accelera- 
ted motion of the train? 

It ought now to be clear that the graph must change its slope 
at every intermediate point as well as at the few points that we 
located, ij it is to represent properly the uniformly increasing velocity 
of the train. 

If we should locate the points for the intermediate half seconds, 
in addition to the points already placed, thus reducing our time 
interval to 0.5 of its former value, and if we should connect all 
the points successively as before, would the broken line thus ob- 
tained more nearly fulfill the condition of changing its slope at 
every point? 

Suppose now that we were to reduce the time interval to 0.2 sec 
and to plot the corresponding broken line (Fig. 8) ; would this line 
approach more nearly than did the other to the line that would 
represent exactly the uniformly increasing velocity of the train? 
It must be clear that by continually diminishing our time intervals 
we shall get broken lines that more and more nearly fulfill the con- 
dition of changing slope at every point, and thus more and 



Fia. 8 
Graph for Fifths of a Second 

more nearly approach to the graph that we want. It is obvious, 

however, that, in a practical problem, it is useless either to ca/rry 

the subdivision of the time inter- 

vol beyond the point at which 

the difference between the broken 

line and a smooth curve is no 

longer perceptiUe in the drawing, 

or to use smaller time intervals 

than we are able to measure by 

means of the timepiece used in 

making our observations. 

In general when we wish to 
make a graph that corresponds 
to a series of observations, we 
locate the points corresponding 
to each of these observations, and 
then draw the smooth curve that most nearly passes through all of 
the points, 

13. Slope of a Curved Graph. As long as \he line is a broken 
one, the slope of the portion between any two consecutive points is 
that of the straight line joining those points; but when we pass to 
a graph that changes its slope at every point, it must be evident 
that the slope at any point is approximately that of a straight line 
joining the given point with a nearby point. The nearer we take 
this point to the given one, the more nearly does the slope of the 
straight line represent that of the curve. 

Since, for this graph, the ordinates and the abscissas represent 
respectively distances and corresponding times, just as they did in 
the graphs for uniform motion, the slope at any point of this graph 
must represent the velocity at the corresponding instant of tinier just 
as it did in their case. 

14. The Train is Stopping. In order to construct the graph 
that will represent the relation between distance and time when 
the train is slowing down, we must again calculate the distances 
of the- train, at the ends of the successive seconds, from the j)oint 



at which the engineer applies the brakes. The velocity at 
this instant is the initial velocity and is 2500 ^ (c/. Art. 8). 
As the speed is decreasing, the acceleration is negative; and so 
(ef. Art. 8), its value is a = - 100 ^2- Hence the final velocity 

for the first second is F = 
and the average velocity v 

Vo-\- at = 2500 - 100 X 1 = 2400 

2500 + 2400 

= 2450^. Multiply. 

ing the average velocity by the time as before, we get for the distance 
traversed in the first second, I = vt = 2450 X 1 = 2450 cm. like- 
wise for the second second we get V = 2300, v = 2400; so that 
/ = 4800 cm. The values for succeeding seconds are as follows: 
sec cm sec cm 

5 11 250 

1 2450 6 13 200 

2 4800 7 15 050 

3 7050 8 16 800 

4 9200 etc. etc. 

15. Graph for Negative Acceleration. Choosing scales such 
that for the abscissas 1 cm represents 10 sec, and for the ordinates 
1 cm represents 10,000 cm, and plotting precisely as before, we obtain 

the graph shown in Fig. 9. In what way 
is this graph for the case of negative ac- 
celeration like that for the case of positive 
acceleration (Fig. 8)? In what way do 
these graphs differ? At the end of what 
second does the train come to rest? As- 
suming that the train then remains at 
rest, add to the diagram the points corre- 
sponding to the next five seconds. At 
the end of what second does the graph be- 
come parallel with the axis of abscissas? 
What, then, is the slope at the end of the 25th second? At the end 
of the 28th? the 30th? What velocity is represented by a slope of 

16. The Entire Motion Bepresented. We have now the 
graphs for the uniform motion of the express train going at full 

ZO ZT> 36 

Fig. 9 
The Train is Stopping 



speed, and for the uniformly accelerated motion with positive and 
negative accelerations while getting up speed and slowing down. 
In order that all of these motions may be represented by a single 
diagram that will go on a page, a 
smaller scale must be used. The 
complete graph appears in Fig. 10 
(1 cm = 30 sec, 1cm =30,000 cm). 
Describe in succession the changes 
of slope. 








17. Equations for TTniformly 
Accelerated Motion. Passing now 
to the anal3i;ical method of repre- 
senting uniformly accelerated mo- 
tion, let us develop an algebraic 
expression that will generalize the 
calculations of Art. 11 and Art. 14. 
If Vq represent the initial velocity, a 

. , 1 i' a *u *• T7 J.U Fig. 10. The Complete Graph 

the acceleration, t the time, V the 

final velocity, and / the distance, then by equation (2), Art. 10, 

V = ^0 + at 
Also, the average velocity, v is found by taking half the sum of the 

6 9 lO 

initial and final velocities; therefore, v = 

^0 + (^0 + «0 


On multiplying this average velocity by the time t to get the dis- 
tance /. we have 

l = Vot + — . 


Equations (2) and (3) are the equations for uniformly accelerated 

The laws of uniformly accelerated motion expressed by 
these equations may be stated as follows: 

1. The final velocity is equal to the initial velocity plus the 
product of the acceleration and the time, 

2. The total distance traversed is equal to the product of the 
initial velocity and the time, plus half tJie product of the accelera- 
tion and the square of the time. 



18. When the Moving Body Starts from Eest. In the cases 
thus far considered the initial velocity was zero. On substituting 
this value in the general expression, the term involving Vq vanishes 

and the equations become V = at and / 

— , which 



relations when the moving body has started from rest. 

It should not be forgotten that when the velocity is decreasing y a, 
the acceleration , is negative. 

19. Acceleration is Not Necessarily Uniform. Throughout 
the preceding discussion we have assumed that the acceleration of 
the train was constant. In reality the case is not quite so simple, 
because the engineer at first puts on the steam pressure gradually, 
and because the acceleration is diminished by the resistance of the 
air, which increases very rapidly when the speed is increased. The 
acceleration which we assumed to be uniform was the average 
acceleration during the time considered. 

20. Determination of Acceleration. The actual experiment of 
determining acceleration is made by observing distances and 

corresponding times, sub- 
stituting their values in 
equation (3), and solving 
for a. 

21. Translatory and 
Eotary Motions. Thus 
far we have considered 
only motion in a straight 
line. We are now ready 
to define motion in general, 
and to distinguish be- 
tween translatory and ro- 
tary motion. 
Fig. 11. Translation and Rotation p^ y^^^^, j^ ^.^y ^^ ^^ j^ 

MOTION with reference to a given point when it is changing either its 
distance or its direction from that point. 


When a rigid body moves in such a way that all its p)oints describe 
equal and parallel paths, its motion is called translation. 

When the motion of a body is such that its points describe cir- 
cumferences about some point or line, its motion is called rotation. 
The point or the line about which the body rotates is called the 
center or the axis of rotation. The planes in which the particles 
move are all parallel to one another, and the axis is necessarily per- 
pendicular to these parallel planes. 

A sled going down a hill has translatory motion only, provided 
there are no turns in the road; for then all of its points describe 
equal and parallel paths. The same is true of a sail boat when it 
is making a straight course. On the other hand, the buzz saw and 
the grindstone are familiar examples of bodies that have rotary 
motion only. Every point on the grindstone, for example, de- 
scribes a circle about a point in the center of the axle on which the 
stone is mounted. The centers of all the circles described by the 
points lie on a line which is perpendicular to the planes of all the 
circles. When an automobile is traveling along a straight road, 
the body of the car has translatory motion only, while the wheels, 
considered with respect to their axles, have rotary motion only; 
but the wheels have both translation and rotation with reference 
to a point on the road. 


1. The units of length and of time are the centimeter and 
the second. Their symbols are cm and sec. 

2. Motion may be either translatory or rotary. 

3. Linear velocity is the rate of change of distance in a given 

4. Uniform linear velocity is measured by the distance traversed 
in one second. Its symbol is g^. 

5. Acceleration is the rate of change of velocity. 

6. Uniform linear acceleration is measured by the change of 
velocity in one second. Its symbol is ^. 

7. Acceleration may be either positive or negative. 

8. The distance traversed by a body having uniformly acceler- 
ated motion is foimd by multiplying the average velocity by the time. 



9. The three methods of representing these relations are: 



(Equation 1) 

V = Vo+ at 
(Equation 2) 


(Equation 3) 

Uniform or average velocity 
equals distance divided by time. 

With uniform acceleration, final 
velocity equals initial velocity 
plus acceleration multiplied by 

Distance traversed with uniform 
acceleration equals initial veloc- 
ity multiplied by time plus half 
the acceleration multiplied by 
time squared. 



1. Define the scientific unit of length, and give its symbol. Define 
the unit of time and give its symbol. 

2. Define the term, linear velocity. What is meant by a constant 
linear velocity? What two things must be stated in order that the 
velocity of a movmg body may be fully described? 

3. How is the numerical value of a imiform velocity found? What 
is the unit of velocity? What is its symbol? 

4. ' Explain how to represent a constant linear velocity by the graph- 
ical method. In connection with the diagram, point out and name the 
co6rdinate axes, the coordinates, and the origin. 

5. What characteristic of the motion of a body is shown by the 
slope of the graph that represents it? 

6. When the velocity of a moving body is changing, how can we 
express numerically the velocity that it has at any instant? 

7. Define acceleration, and illustrate by a numerical example. 

8. When is an acceleration positive, and when negative? What is 
meant by uniformly accelerated motion? 

9. Draw the graphs that represent the relation of distance to time 
for a positive and a negative acceleration. In these graphs, what does 
the slope represent? 


10. What changes of slope occur in the graph when the acceleration 
is positive? When the acceleration is zero? When it is negative? 

11. When a graph is curved, what line will represent approximately 
the direction of its slope at any point? 

12. Define motion, and distinguish between translatory and rotary 
motion, illustrating by examples. 


Note. 1 m = 100 cm = 39.37 inches. 

1. A runner passes over 100 yards in 10 sec; what is his speed 
cm p 


2. What is the speed of a race horse that covers a mile in 2 min- 

utesWin^^ (2) in 52? 

3. What is the speed of an automobile that runs a mile in 55 sec. 
W<^^ (2)i„|E? 

4. Sound, at 0° Centigrade, travels 1090 feet in one sec. What is 
the speed of sound in ^? How many seconds would it take to trav- 
erse 1000 m? 

5. What was the average speed of a railroad train that traveled 
134 mUes in 115 minutes (1) in 2^|~? (2) in ^? 

6. Express the velocity of 1 ^^ in ^*, and in ^. 

7. A sled, started from rest and going down a hill of uniform slope, 
traverses 900 cm from the starting point in 3 sec. What is the accelera- 
tion, and what the final velocity? 

8. A wagon starts down a hill with a velocity of 30 — and its 

acceleration down-hill is 80 — ^? What is its velocity at the end of 5 
sec? What is the total distance traversed in the same time? 

9. A wheelman, starting from rest, had attained at the ends of 
the first three seconds the following distances: 1 sec, 90 cm; 2 sec, 
360 cm; 3 sec, 810 cm. Supposing the acceleration to remain con- 
stant during that time, what is (1) the acceleration? (2) the velocity 
at the end of 6 sec? (3) the distance traversed at the end of 5 sec? 
(4) the distance traversed during the 5th sec? [For (4), subtract 
the distance attained in 4 sec from that attained in 5 sec] 

10. The results of experiments show that the acceleration of a body 
allowed to fall freely is 980 -^j- (") Calculate the distances attained 
by the falling body when given an initial velocity of 10 cm vertically 
downward, making a table of distances and times up to 10 sec. (b) 
Choosing a convenient scale, plot a graph representing the motion. 

11. Calculate the velocities of the falling body for the times given 
In problem 10. 



1. Which of you can make the longest list of the motions with 
which you are familiar, classified under the headings: Uniform, 
Uniformly Accelerated Positive, Uniformly Accelerated Negative, 
Translatory, Rotary? 

2. Mark your height on a door-post; measure it in inches and. in 
centimeters. From these measurements can you find out how many 
centimeters are contained in one inch? 

3. In ypur debating society, choose for a question the following: 
Resolved: That the general adoption of the metric system of weights 
and measures is advisable. For data write to the National Bureau of 
Standards, Washington, D. C. 

4. Which of you can find out the most interesting facts about 
Galileo and his knowledge of falling bodies? 


22. The Production of Acceleration. In the preceding chap- 
ter, we attempted to get clear notions about uniform and accelerated 
motions, without considering the factors upon which their production 
and variation depend. What are the relations that determine 
whether the motion shall be uniform or accelerated ? What relations 
determine the amount of the acceleration? We can most easily find 
the answers to these questions by again studying the train. 

Let us first suppose that the train drawn by the engine consists of 
six cars all alike. Let us also suppose that the engine, using its full 
power, can impart to this train an acceleration of 50 |^. Now, if 
this engine be replaced by a smaller one having less power, will the 
acceleration that this smaller engine can impart to the same train be 
greater or less than 50 ^? Must the engine that can impart to this 

train an acceleration of 60 ^^^2 have greater or less power than the 
first engine? 

Those who can not answer these questions from observations 
made upon the train itself, will readily answer them by inference from 
similar cases. Thus, everybody knows that more force is required 
for imparting to a ball a great velocity in a given time than 
for imparting to it a small velocity in the same time, that two oarsmen 
can impart to a boat a greater velocity in a given time than can one, 
that greater effort is required by a bicyclist to attain a great velocity 
in a given time than to attain a small velocity in the same time. 
Observation and experience lead us habitually to associate a 
greater acceleration with a greater effort or force. 

23. Acceleration and Force. Although common experience 
gives us this general information, it does not give us the specific 
numerical relations. This information can be obtained only by 
making careful measurements of the quantities involved. 




Thus, if we measure the pulls of different sized engines, having 
different powers, and observe the corresponding accelerations, 
arid if we make proper corrections for friction of the moving parts 
and for air resistance, we shall find that the numbers representing 
the pulls are directly proportional to the numbers representing 
the accelerations imparted to the train. 

Many experiments of this sort have been devised and carried 
out in physical laboratories to test the validity of this conclusion, 
and they all tend to establish the truth of the general principle that 
when different accelerations are given to the same body, the ratio of 
the numbers by which we express the forces to those by which we 
express the corresponding accelerations is constant. 

Fig. 12. Eight-Oared Shell 

Another illustration will help to make this clear. When only 
two of the crew of an eight-oared shell row, they can impart to the 
boat a certain acceleration. After making proper allowance for the 
increased resistance of the water and air, it will be found that when 
four row, they can impart an acceleration twice as great, six an 
acceleration three times as great, and so on. 

24. Different Bodies Having the Same Acceleration. We have 
thus far considered how the forces vary when different accelerations 
are given to the same body. Let us now consider how the forces 
vary when the same acceleration is given to different bodies. 

If an acceleration of say 50 ^ can be given by a certain engine 
to a train of five empty cars, must the engine that can give the 
same acceleration to a train of ten similar cars be more or less 
powerful than the first? Again, if the same acceleration is to be 



given to the train of five cars loaded with passengers, can the same 
engine do the work? 

Common experience again gives us qualitative answers; for 
everybody knows that an engine that can easily move a short 
train may fail to move a long one, so that another engine must 
be added. Likewise it re- 
quires greater effort on the 
part of a bowler to give a 
large ball a certain velocity 
than to give a small ball 
the same velocity in a given 
time; and it requires the 
efforts of more oarsmen to 
give a certain acceleration to 
a big boat than to a little one. 
The student can recall many 
similar facts from his daily 

It appears, then, that the more we increase the size of a body, 
the substance remaining the same, the greater is the force required 
to give it a certain acceleration. 

Fig. 13. Small Engine: Short Train 

25. Mass. In order to get quantitative relations, experiment 
is necessary. If we measure the pull of an engine when it is im- 
parting an acceleration of 50 ^ to a train of five empty cars, and 
again when it is imparting the same acceleration to a train of ten 
similar cars, we shall find that the pull in the second case is twice that 
in the first. Likewise we shall find that the pull for a train of fifteen 
6ars is three times that for five cars, and so on; i.e., when the 
acceleration is the same, the numbers representing the forces are 
directly proportional to the corresponding numbers of cars. 

The matter appears very simple as long as the cars are empty 
and all alike. But although we know from experience that more 
force is required to impart a given acceleration to a loaded 
train than to an empty one, yet it is impossible to determine how 
much force, until we have adopted a means of comparing the loaded 
train with the empty one. 


These differences in the make-up of the trains, whether in the 
number of cars or in the load, are differences in mass. 

26. Masses Compared by Forces. It is easy to see that when 
two trains consist of precisely similar cars, all of them empty, 
the train of ten cars has twice the mass of the train of five cars, be- 
cause it is made up of just twice as many units of the same kind. 
But it has just been shown that to impart a certain acceleration to a 
train of ten empty cars the force is twice as great as that for a train 
of five empty cars; so that we may compare the masses of the two 
trains not only by the numbers of cars, but also by the forces re- 
quired to give them the same acceleration. 

When the differences in the trains are differences in the loads, 
we can not compare their masses by comparing the number of 
cars, because the units are not alike. Therefore we must resort 
to the other method, that of comparing the masses by the forces 
that can impart to them the same acceleration. This method is 
applicable to all bodies, whether composed of like or unlike kinds 
of matter. Therefore, in general, tvx) masses are equal when, under 
the same conditions, equal forces can impart to them equal accelera- 

Applying this method to the cars, it appears that when an engine 
can give two empty cars the same acceleration that it can give to a 
single loaded car, the combined mass of the single car and its load 
is equal to that of two empty cars ; and therefore in this case the mass 
of the load is equal to the mass of one of the cars. 

For a given acceleration, then, since the masses are equal when 
the forces are equal, it follows that if one of the masses be doubled, 
the corresponding force is doubled ; if the mass is made thrt^ times 
as great, the force is also three times as great; and so on. In general, 
then, when the acceleration is constant, the forces are proportional 
to the masses. 

27. Force, Mass, Acceleration. Since we have shown in the 
preceding paragraphs that when the mass is constant, the force 
varies directly as the acceleration, and also that when the acceleration 
is constant, the force varies directly as the mass, it follows that, in 



general, the force must vary directly as the product of the mass and 
the acceleration. If we choose our units of force appropriately, 
and if we let / represent the force, m the mass, and a the acceleration, 
we may write 

/ = ma (4) 

This equation defines the force in terms of mass and acceleration. 

In connection with this equation, it is to be noted that if m 
and / are constant, a must be constant also; i.e., if a body be 
acted on by a single or an unbalanced constant force, its motion 
will be uniformly accelerated. 

Besides magnitude, every force has two other characteristics, 
namely, its direction, and its point of application. When these 
three characteristics are specified, the force is fully described. 

28. Unit Mass. Thus far we have taken an empty car as 
the unit of mass; but it is manifest that accurate measurement 
necessitates the establishment of a 
unit that is fixed and at the same 
time more convenient. Therefore, 
just as we have a standard of 
length, the meter, we have also a 
standard of mass. The interna- 
tional STANDARD OF MASS is a cer- 
tain piece of platinum which is 
carefully preserved at Paris along 
with the standard meter, and is 
called the kilogram. The unit of 
mass employed by all scientists is 
the gram, which is the one-thou- fio. 14. 
sandth part of the mass of the 
standard kilogram. The abbreviation for gram is gm. To express 
very large masses, the kilogram is a more convenient unit. The 
abbreviation for kilogram is Kg. 

Smce we are now able to express both mass and acceleration in 
terms of grams, centimeters, and seconds, we may also express force 
in terms of these same fundamental units. Thus in the equation 

Standard Kilogram: 
Actual Size 


/ = ma, if we substitute m = 1 gm, and « = 1 ^2> ^'^ obtain 
/ = 1 X 1 = 1 ^^^> i^7^ic^ defines the scientific unit of force as thai 
force which can impart an acceleration of one centimeter per second 
per second to a mass of one gram. This unit of force is called the 
DYNE. Note that the number of units of force is obtained by 
multiplying together the numbers representing mass and accelera- 
tion. This operation gives us gm x ^2- Hence the symbol for 
the dyne is ^^. 


By definition (c/. Art. 10), a = — - — ^, therefore f ^ ma = 
The product m{V — v^), or mass X change of veloc- 


711 ( V — V ^ 

ity, is called change of momentum; — ^^ — - — — is therefore the 

rate of change of momentum; and since it is equal numerically to 
may it is also a measure of the force to which it corresponds. 

29. Weight. Since we have learned to state the relation 
of force to mass and acceleration, we are in a position to get 
some definite ideas concerning a subject about which there are 
many common misconceptions. We all learned in early childhood 
that bodies, including ourselves, fall to the earth when unsup- 
ported. We are accustomed to associate this motion with a force 
called gravity, which we conceive acts so as to attract all bodies 
toward the earth. The attraction between the earth and any par- 
ticular body is called its weight, and tends to give the body an 
acceleration vertically downward. This fact is also a familiar one, 
for everybody knows that a body falling from a great height 
acquires a greater velocity than does one falling from a less height. 
It is our knowledge of this fact, acquired from very early expe- 
rience, that impels us to avoid a high fall. 

Now, what is the relation between the weights of bodies and 
their masses? Equation (4) will give us the answer. Thus^ 
the weight /, in dynes, of any body whose mass is m, is equal to 
this mass multiplied by a, the acceleration that this weight will 
give it if it is allowed to fall freely; i.e., / = ma. Similarly, the 
weight f of any other body having a mass m', and receiving an 
acceleration a\ is /' = m'a'. In order to find the ratio of the two 



weights in terms of their corresponding masses, we must divide one 
of these equations by the other, thus: j = . We therefore see 


that if both bodies have the same acceleration when falling 
freely, i.e., if o' = o, then their weights are proportional to their 

30. Gktlileo's Experiment. The question to be answered 
now is, When two bodies have different masses, does the attrac- 
tion of the earth give them equal accelerations? From the time 
of Aristotle to the end of the 
sixteenth century, this was a 
much disputed question. Aris- 
totle (384-322 B.C.) taught 
that if t\/o bodies of unequal 
.mass were dropped from the 
same height at the same in- 
stant, the heavier body would 
reach the earth first; and his 
followers defended this opinion 
by his authority and by argu- 
ments based upon what they 
thought ought to be the nature 
of things. Galileo (1564-1642 
A.D.) was the first to recognize 
that svxih a dispute can be set- 
tled only by experiment. Ac- 
cordingly, about the year 1590, 
he performed the experiment 
of dropping at the same instant 
a small cannon ball and a large 
bomb from the top of the Leaning Tower of Pisa. They reached 
the ground at very nearly the same instant; so he came to the con- 
clusion that if it were not for the resistance of the air, they would 
have fallen in exactly the same time. The fact still remained, how- 
ever, that a body with a large surface in proportion to its mass, 
such as a feather, was known to fall very much more slowly than 

Fig. 16. Leaning Toweb op Pisa 


a piece of metal. After the invention of the air pump, in 1660, 
it became possible to settle the dispute finally. This was done 
by showing that when a feather and a coin were dropped simul- 
taneously in a long tube from which the air had been pumped, 
they fell side by side and reached the bottom at the same time. 

31. The Belation between Weight and Mass. Reasoning from 
these experiments by means of equation (3), Art. 17, it follows 
that the accelerations of all freely falling bodies are equal. For 
if two bodies fall simultaneously through a distance I in time <, 
then for the first, since the weight and hence the acceleration is 

constant, I = -—; and likewise for the second, V = —^, But 

since the distance is the same for each, as is also the time, I = V 
and t = t', whence a = a'. 

Thus it has been proved that at any given place, the accelera- 
tion due to the earth's attraction is the same for all bodies and that 
therefore, so long as they are compared at the same place, tlie 
weights of all bodies are proportional to their masses, 

Galileo, wishing to prove this statement with greater accuracy, 
devised experiments with pendulums of different mass. These 
experiments verified more accurately the same conclusion. Re- 
peated with greater refinement by Sir Isaac Newton and others, 
they have given convincing evidence of the truth of this statement. 

From what has just been stated, it follows that v)e can cmtt- 
pare vmsses by comparing their weights. This is the method in 
common use; but it must be noted that, since the attraction of 
the earth for a given body is different at different places, the 
weights of the two masses that are to be compared must in gen- 
eral be determined at the same place. For the comparison of 
masses by means of their weights, the equal arm balance is gen- 
erally used. 

32. Density. In connection with the masses of different bodies, 
we have seen that bodies having equal volume may differ greatly 
in mass. Thus, one cubic centimeter of lead has a much greater 
mass than has one cubic centimeter of water; while the latter has 
a greater mass than has one cubic centimeter of wood. The 


appropriate measure of the density of any substance is the mass 
in unit volume at a temperature of zero degrees Centigrade. Thus, 
if the mass of a specimen of a certain kind of glass is found to be 
25 gm, and its volume 10 cm', the average density of the glass is 
. 1 of 25, or 2 . 5 grams per cubic centimeter. 

If D represent the density, m the mass, and V the volume, 
these relations are stated analytically by the equation 

D = - 

As defined by this equation, the density of a substance is its 
mass per unit volume. The unit of density is one gram per cubic 
centimeter, and its symbol is ^^, Since the gram was intended to 
be the mass of 1 cm' of water, and since it is so, very nearly, the 
number of cm' in the volume of a quantity of water is the same 
as the number of gm in its mass. The density of water, there- 
fore, may be taken as 1 ^^. 

33, Work. •, In Chapter I we have studied the motion of a rail- 
road train and seen how that motion is produced by the engine. 
Why does the engine move at allf Must more steam be used to move 
a train of large mass than to move a train of small mass? Must 
more steam be used to move a given train over a long distance 
than over a short one? Other conditions being the same, does 
it require a larger amount of coal to generate a larger amount of 
steam? The student probably knows the answers to these ques- p 
tions, and also in a general way that, other things being equal, the 
amount of coal required is proportional to the amount of work to 
be done. Since most kinds of work, like that done by the locomotive, 
consist in putting bodies into motion, and in maintaining tlieir 
motions in opposition to resistances of some sort, and since some- 
body always has to pay for getting work done, it becomes 
necessary to know definitely just what an amount of work depends 
on, and to have a unit in terms of which all kinds of work may be 

34. Work, Force, Distance. If an engine or a horse or a 
man is doing any kind of work it is evident that, other thuigs being 



Fig. 16. Plowing 
Work is proportional to force and to distance. 

equal, the amount of work done is directly proportional to the push 
or pull, i.e., to the force exerted by the agent that does the work. 
Thus, if each of two engines pulls its train on a straight and level 
track for the distance of a mile, and if the second engine has to pull 
with twice the force of the first, it is clear that the second engine 

must do twice the 
work that the first 
does. Again, suppose 
that the second en- 
gine pulls for . one 
mile, and then con- 
tinues to pull with 
the same force for 
another mile, it must 
again be clear that in 
pulling the train two 
miles it does twice 
the work that it did in pulling it one mile. It follows, then, 
that if the second engine, exerting twice the force of the first, and 
hence doing twice as much work per mile as does the first, should 
continue pulling with this force through a distance of two miles, 
it would do four times as much work as the first engine did in 
pulling its train one mile. 

Since, then, the amount of work done by an agent is directly 
proportional to the force and also to the distance through which 
the agent acts, and since the amount of work depends on these two 
factors only, it follows that when the units are properly chosen, 
the measure of the work done is the jyroduct of the numbers repre- 
senting the farce and the distance. In symbols, if / represent the 
force of the agent, and I the distance through which it acts, and if 
W represent the work done, then 

W = fl. (5) 

This is the equation for work. 

35. TTnitWork. The unit in terms of which work is meas- 
ured may easily be defined with the help of the equation W = //, 
for if / = 1 dyne and 1=1 cm, we have W =1x1 = 1. 



Therefore, since the equation gives unity for the work when 
the force is one unit and the distance one unit, it is most con- 
venient to define the unit of work as the amount of work that is 
done when a force of one dyne acts through one centimeter. This 
is the unit of work adopted by physicists, and it is called the erg. 
Since the symbol for the dyne is ^^^, and since the number 
of ergs is obtained by multiplying together the number of dynes 
and the number of centimeters, it follows that the symbol for the 

36; Unergy. We now come to the question of the relation 
betweeWlhe^ amount of coal burned and the amount of work done. 
It is generally recognized that 
a water wheel, in order to 
move machinery continuous- 
ly, must be continuously sup- 
plied with water, which must 
be allowed to fall upon it 
from a higher level; that a 
windmill will not continue 
to pump water unless the 
wind continues to blow 
against its blades, that a 
horse or a man can not con- 
tinue to do work unless he 
regularly consumes food, and 
that an engine of any sort 
must continuously consume 

coal in order that the steam may be kept up at the necessary pres- 
sure while it is doing its work. 

For centuries the most careful thought of philosophers and the 
greatest genius of inventors were employed in trying to think out 
and construct some device for obtaining perpetual motion, i.e., a 
device which would continue to move indefinitely without a con- 
tinuous external supply of energy. Since every such attempt has 
been unsuccessful, scientists have become convinced that a per- 
petual MOTION MACHINE is impossible. Thus, if any machine be 

Fig. 17. Haying 
A man can not work unless he consumes food. 



at rest, it can not start itself; and if it be in motion, the greater 
the friction of its moving parts the sooner will it stop. If it be 
harnessed to other bodies and made to do work in moving them, 

it will come to rest all the sooner. It 
can be made to work continuously 
only by supplying it continuously with 
ENERGY from some external source. 
Energy y then, represents ability to do 
work. In the case of the water wheel 
the energy is derived from the motion 
of a mass of water; in the case of the 
windmill, from the motion of a mass of 
air; in the case of the horse or man, 
from the consumption of a quantity of 
food; and in the case of a steam or gas 
engine, from the consumption of a 
quantity of fuel. 

Thus it becomes evident that to 
do work energy must be expended, and that to store up this energy 
work o{ some sort must have been done. Now, many careful 
experiments with all forms of energy have shown that a given 
amount of energy always corresponds to the same amount of 
work, whether that energy be expended in doing the work, or 
the work be done in storing the energy. 

Fig. 18. The Windmill 

It will not go when there is no 


37. Energy Measured by Work. Since the energy of a body 
is equivalent to the work it can do, and also to the amount that had 
to be done an it in order to impart the energy to it, we may measure 
this energy by measuring either of these amounts of work. Some- 
times one of these methods is more convenient, sometimes the other. 
For example, let us consider the energy necessary to run an eight- 
day clock. Such a clock is usually operated by a spiral spring, or 
by a weight which is raised by winding up the cord upon which it 
hangs. Suppose that the weight has a mass of 5000 gm. Then, 
since / = ina, the force with which it pulls on the cord is ma, 
or 5000 gm multiplied by the acceleration that it would have if 
allowed to fall freely in consequence of the earth's attraction. This 



acceleration, which we have learned is the same for all bodies, is 
found by experiment to be, at sea level and in the latitude of New 
York, 980^. Therefore, the force with which we must pull in 
order to lift this mass is 5000 X 980 = 4,900,000 dynes. To avoid 
the repetition of zeros it is convenient to write this 49 X 10^. 

If the distance through which the mass is lifted is 100 cm, then 
from Art. 34, W = fl = 4 900,000 X 100 = 49 X 10^ ergs. Since 
this is the amount of work done in 
winding up the clock, it represents 
the energy stored in the lifted weight. 
Likewise when the weight descends, 
it does work in running the clock and 
this work is again f X I = 5000 X 
980 X 100 = 49 X 10^ ergs. Since 
this is the work done by the energy 
stored in the lifted weight, it also is 
a measure of that energy. Thus, in 
general, if we can measure or com- 
pute the work done on a body in 
imparting energy to it, or the work 
that it does when it parts with its en- 
ergy, we can determine the amount of 
energy that it had. 

It will be noted that in the exam- 
ple just given a small amount of use- 
less WORK, done in overcoming fric- 
tion while winding the clock, was 
neglected. In every case in which 
energy is transformed or transferred some of this useless work is 
done. If the amount of useless work is at all comparable with that 
of the useful work, allowance must be made for it. The ratio of the 
useful work done to the total amount of energy expended is called 
the EFFICIENCY of the machine by which the transformation or 
transference i$ accomplished. 

Suppose now that in the clock just considered we were to replace 
the weight by a spiral spring. How much energy must the spring 
have when wound up in order that it may be able to run the clock 

Fig. 19. Hoisting Coal 
Work equals force times distance. 



for as long a time as the weight ran it? How much work would 
have to be done in winding up the spring? 

38. Energy is Potential or Kinetic. In the cases that we 
have considered, energy has been stored in a lifted weight, a 

coiled spring, unbumed coal, 
and unconsumed food. En- 
ergy of this kind that a body 
has because of its position or 
internal condition, so that it 
tends to move and do work, is 
called potential energy. In 
the case -of the windmill or 
the water wheel, the energy 
of the air or water is due to 
the fact that it is in motion. 
Likewise a base ball or can- 
non ball does work while it 
is being stopped. Hence it 
also possesses energy; and it 
must be quite clear that it 
has this energy because of its 
motion. The energy that a moving body has because of its motion 
is called kinetic energy. 







Fig. 20. Pile Driver 

The weight has potential energjr when it is 

raised, kinetic when it strikes. 

39. The Kinetic Energy of a moving body being evidently 
due to its mass and velocity, it is often more convenient to measure 
it in terms of these quantities than in terms of work done. This 
may readily be done with the help of equations (3), (4), and (5). 
Thus W = fl, in which / is the average force used in imparting 
to the moving mass its velocity, and I the distance through which 
this force acted. Also this force / = ma, in which m is the mass 
of the body, and a its acceleration while acquiring its full velocity. 
Therefore the work done in giving the body its kinetic energy is 
W= fl = mal. Again, by equation (3), / = ^af, and by equation 
(2) V = at, in which I is the distance traversed in the time t while 
acquiring the velocity V with an acceleration a. Since we do not 


care to know the time t, we may eliminate it by substitution. Thus, 

V P 

from (2) / = — ; then f = —j. Substituting this value for t in 

equation (3), we have I = -^ = — . Finally, by substituting this 
value for I in the equation W = mat,, we find that the energy is 
e = TF = — ^ — . Simplifying, we have 

c == — - ergs. (6) 

Since the symbol for mass is gm, and that for velocity is ^, 
the symbol for kinetic energy is ^^^^ . Note that this is the same 
symbol as that for the unit of work, as it should be, because en- 
ergy is measured by work. 

The advantage of deriving the equation e = --r— is mani- 
fest when we apply it to the case of throwing a ball or firing a shot. 
For while it would be very difficult to measure t and a correspond- 
, ing to the distance I through which the force of the hand or the 
powder was exerted, it is not so very difficult to measure F. Hence 
it is often desirable to have an equation in which a and t are not in- 
volved. In many cases, however, the kinetic energy of a body can 
be measured with convenience by the work that it does when it gives 
up its energy in stopping, 

40. Newton's Laws of Motion. From all that has been said 
it must be apparent that a body can not of itself start, or stop, 
or otherwise change either the rat6 or the direction of its motion. 
This fact is often expressed by saying that every body has inertia. 

The relations of the phenomena with which we have become 
familiar in this chapter were described tersely by Sir Isaac Newton 
in the following statements, which first appeared in his celebrated 
Principia in 1687. They are known as Newton's Laws of Motion. 

1. Every body continues in its state of rest or of uniform motion 
in a straight line, except in so far as it is compelled by force to 
change that state. 

2. Change of motion is proportional to the force impressed 


and takes place in the direction of the straight line in which the 
force acts. 

3. To every action, there is always an equal and contrary re- 
action, or the mutual actions of any two bodies are always equal 
and oppositely directed. 

41. Illustrations. All these laws are illustrated in the train. 
The train can not start unless pulled by the engine, which exerts 
force upon it, i.e., imparts energy to it. Once started, the train 
can not stop itself. If brought to rest it gives up its energy in over- 
coming the resistance of the air, the friction of the moving parts, 
and the friction of the brakes when they are applied to the wheels. 
If there were no such resistances, the train, once set in motion 
with a certain velocity, would continue to move without change 
either of speed or direction. Again, while starting, if there were 
no friction, and if a constant force were applied, there would be an 
increase of velocity the same for each second, i.e., the rate of change 
of motion, as measured by the product of the mass and the accelera- 
tion, is proportional to the force; and it is in the direction of the 
straight track along which the engine pulls. 

When the brakes are applied, their force, and therefore the 
corresponding acceleration, is in a direction opposite to that of the 
motion; and if this force remains constant, there is a decrease of 
velocity the same for each second. Here again the total change of 
motion is proportional to the force impressed and takes place in 
the direction in which this force acts. 

But why is it that the train when under full head of steam does 
not continue with uniformly accelerated motion, and therefore 
increase its speed indefinitely, instead of reaching a certain speed, 
which it can not surpass? The answer is that the resistance of the 
air increases very rapidly, and therefore the engine soon has to use 
all its energy against air resistance and internal friction ; so that there 
is no excess left to do the work of increasing the motion of the train. 
The total external force opposed to the motion of the train is 
exactly equal to the total external force urging it forward; and 
therefore the result is the same as if no force were a^cting at all — 
namely uniform motion in a straight line. 



Fig. 21. Diving 

The boat has an acceleration in the opposite 


Now where are we to look for the application of the third law? 
We have seen that the engine can not move the train if the driving 
ivheels sHp; therefore it appears that the force of the engine is 
applied at the place where the drivers bear upon the track. The 
engine, then, tends to push the track backward, and would do so 
if the track were free to 
move. But the track is 
made fast to the earth, 
and therefore the engine 
tends to push the whole 
earth backward. The 
force of the engine, the 
action, is equal to ma, 
i.e., to the total mass of 
the engine and train mul- 
tiplied by the accelera- 
tion that they acquire. 
Also the resistance of 
the earth, the reaction, 
is equal to mV, i.e., to the mass of the whole earth multi- 
plied by the acceleration that it receives. This force and accel- 
eration are oppositely directed with respect to those of the train. 
Why does not the earth move? The answer is that it does move, 
but so little that the motion is imperceptible. This will easily 
be understood when we remember that the two forces are equal, i.e., 
mV= ma. But, dividing both members of the equation by m'a, so 

as to get the ratio of the accelerations, we have —7- = -7-, whence 
° ma m'a 

— — — ;, which tells us that when the forces are equal the accelera- 
a m' 

tions are inversely as the masses. Since the mass of the earth is 

very large compared with that of the train, it is evident that the 

acceleration of the earth must be very small. If the masses were 

more nearly equal, the accelerations would be more nearly equal. 

Every one knows that when he dives or jumps from a small boat 

it has a perceptible acceleration in the direction opposite to that in 

which he jumps; and if he jumps from a larger boat, the accelera- 


tion of the boat is smaller; while if he jumps from a big ship, the 
acceleration of the ship is imperceptible. So it is with the engine 
and the earth. 

42. Kate of Boing Work. In connection with work and 
energy there remains another important question to be considered. 
How are we to measure the rate at which energy is supplied; or, 
what comes to the same thing, how are we to measure the rate 
at which work is done? With the units we have adopted this is 
very simple. We have only to calculate the number of ergs of 
work done per second. Thus, if 120,000,000 ergs are done in 60 
sec, then in 1 sec there will be perfonned one sixtieth of 120,000,000 
or 2,000,000 ergs. The rate, then, is 2,000,000 ^^|. The rate 
at which any agent does work is called its power or activity, and 
the power is measured by the work that it can do in one second ; i.e. 

Power = -^^ . 

43. Engineering Units. For measuring force, work, energy 
and power, engineers use a system of units based on the pound 
weight, the foot, and the second. These units are not nearly so 
convenient as those based upon the centimeter, the gram, and the 
second; but since they are so widely used in engineering practice 
they are here described for reference. Those students who expect 
to prepare themselves for engineering, should master these defini- 
tions and be able to apply them in numerical problems. 

Since the foot is equal to 30.48 cm, the numerical value of the 

980 ft 

acceleration of gravity in this system is q , or 32.2 — g (nearly). 

oU.4o sec 

This quantity is usually denoted by g. 

Instead of deriving their unit of force from a unit of mass and 
a unit of acceleration, as the physicists do, engineers use as their 
unit of force the weight of a pound mass at sea level and in the lati- 
tude of New York, and call it the pound-force. 

Whenever in an engineering equation the mass of a body ap- 
pears — as in the case of kinetic energy — it should be noted that we 
must eliminate it from this equation with the help of equation (4), 


which expresses the relation between the mass of a body and its 
weight. Thus, / = ma, whence m = — . But / is expressed in 

pounds-weight, and o, the acceleration in this case, is 32.2 — -^; 

, . ., p , 1 pounds-weight of the body, 
therefore the mass of a body m = ^^-^ ~ 

which expression must be substituted for the mass in the given 

The amount of work done or of energy expended when a jxmnd- 
force is exerted through the distance of 1 foot, is called one foot-pound. 
By equation (5), Art. 34: 

W (in f oot-pounds)\ = // = pounds-force X feet. 

To get the measure of the kinetic energy of a body, in terms 

of its weight and velocity, we must resort to the equation, 

e = — ^— . Since in engineers' units the mass m of the body is 

~ ^^ — , and since the velocity V is expressed in feet per 

second, the equation becomes^ 

pounds-weight (feet per second)^ 
e = - X 2 ' ^^ 

.... , . pounds-weight X (feet per second)' 
e (m foot-pounds) = ^ 32 2 X 2 

The engineers unit for power or activity is tlte horse-power, 
which is the rate at which work is done or energy expended when 
550 foot-pounds of work are done in ea^h second. Hence 
-J. _ pounds-force X feet 

^ 550 X seconds 

One horse-power is found to be equal to 746 X 10^ ^^. 

On the continent of Europe, engineers use a system of units 
based upon the kilogram, the meter, and the second. These units 
are defined or derived in a similar manner. 

Thus, the kilogram-meter is the work done, or energy 
expended, when a force that is equal to the weight of a kilogram 
mass is exerted through the distance of 1 meter. Hence, 

W (in kilogram-meters) = fl = kilograms-force X meters. 


One kilogram-meter equals 980 X lO"* ergs. 

Since g, the acceleration of a freely falling body, expressed in 
meters and seconds, is 9.8 ^, the equation for kinetic energji in 
kilogram-meters is 

_ mV^_ kilograms-weight (meters per second)' 

^ - "2" - g ~ ^ ] 2 ' ""^ 

,. , ., , V kilofframs-weiffht X (meters per sec*. )* 
e (m kilogram-meters) = . 

To solve problems in which the relations are expressed by these 
equations, it is necessary only to substitute the known values for 
the quantities represented in the equations, each expressed in its 
appropriate units; and then the unknown quantities can be found, 
provided, of course, that in the statement of the problem, one equa- 
tion can be formed for each of the unknown quantities. 


1. To describe a force completely we must state: 1, its point 
of application ; 2, its direction ; 3, its magnitude. 

2. Two bodies are said to have equal masses if equal forces give 
them equal accelerations. 

3. The unit of mass is the gram, and its symbol is gm. 

4. The unit of force is the dyne and its symbol is ^™^. 

5. Force is measured by the product of the mass and the 
acceleration, i.e., / = ma, 

6. If a body that is free to move be affected by an unbalanced 
constant, force, the motion will be uniformly accelerated. 

7. At any given place, all freely-falling bodies have the same 
acceleration. At sea level in the latitude of New York this 
acceleration is 980 ^; therefore, since / = ma, a mass of 1 gm has 
a weight of 1 X 980 = 980 dynes. 

8. At any given place the weights of bodies are proportional 
to their masses; therefore the masses of two bodies may be com- 
pared by comparing their weights. 

9. The density of a substance is its mass per unit volume. 
Its symbol is ^,. 

10. When masses are moved, or when their motions are changed, 


work is done; and the measure of the work done is the product of 
the force and the corresponding displacement, i.e., W = fl, 

11. The unit of work is the erg, and its symbol is ^^^^ . 

12. Scientists are convinced that a perpetual motion machine 
is impossible. 

13. The doing of work implies the transfer of energy, and in 
every such transfer two bodies are equally and oppositely affected. 

14. The energy transferred is measured by the work done, 
i.e., e = W, 

15. The ratio of the useful work done to the total amount of 
energy expended is the efficiency of the machine. 

16. Energy is either potential or kinetic. 

17. Kinetic energy may also be measured in terms of mass and 

velocity, i.e., e = W= -r— . 

18. Activity is the rate of doing work, and is measured by the 
number of ergs done per second. 

19. Engineering units are, for force, the pound-force; for work 
or energy, the foot-pound; and for activity, the horse-power. 


1. In what two ways may bodies differ in mass? 

2. When different forces act on the same mass, what is the relation 
of the forces to the corresponding accelerations? Illustrate by exam- 

3. When the same acceleration is imparted to different bodies, 
what is the relation between the forces and the corresponding masses? 

4. What kind of motion results from the action of a single or an 
unbalanced constant force? 

5. Define the cm-gm-sec unit of force and give its name and sym- 

6. What is meant by the weight of a body? 

7. What is the use of the equal arm balance, and why can it be 
employed for this purpose? 

8. Of what does work consist, and what is the numerical meas- 
ure of an amount of work? Write the equation for work. 

9. Name and define the cm-gm-sec unit of work. Give its symbol. 
10. When is a body said to possess energy? 


11. What is meant by the term perpetual-motion machine? What 
reason have we for believing that no such machine can be made? 

12. What are some of the sources from which our supplies of en- 
ergy ordinarily come? 

13. When a body possesses energy, is all of it available for the 
doing of useful work? Illustrate by some examples. 

14. Define the terms kinetic energy and potential energy, and 
give some examples of each kind. 

15. What is the advantage of having an equation for kinetic energy 
in terms of mass and velocity? 

16. Explain the application of each of Newton's laws to the cases 
of a rimner, a bicyclist, or an automobile. 

17. If all the moving bodies on the earth, such as railroad trains, 
steamships, and animals, were to travel eastward at the same time, 
and continue to do so indefinitely, what would be the ultimate effect 
upon the eastward velocity of the earth's rotation? 

18. Define the engineer's units of force, work, energy, and activity. 

19. What expression should be substituted for mass when engi- 
neer's units are employed? 


1. The masses of two loaded cars are 40,000 and 50,000 lb. re- 
spectively; If a locomotive engine, exerting 2000 pounds-force on the 
first car, gives it an average acceleration of 2.0 — j, what acceleration 
would it give to the second car? What force would give the second 
car the same acceleration as was given the first? Note: Friction is not 
here considered. Each car would require a certain force to overcome 
this, in addition to that required for the acceleration. 

2. Five men, rowing a boat, give it in 30.0 sec a velocity of 15.0 
— . What is the average acceleration? All otlier things remaining 
the same, what velocity would be given the boat by three men? What 
is the average acceleration in this case? What in each case is the dis- 
tance traversed in the 30 sec? 

3. A base ball has a mass of 140 gm, and is thrown from home 
base to first, a distance of 2743 cm, in 0.90 sec. What is its velocity? 
If the catcher applied the throwing force during 0.10 sec, what was 
the average acceleration during that time? What was the force in 
dynes? If it was stopped by the first baseman in 0.05 sec, what then 
was the amount and sign of the acceleration? What force did it exert 
on his hands? 

4. What velocity and acceleration are given to a mass of 500.0 gm 
by a force of 50,000 dynes applied for 10.00 sec? Tlirough what dis- 
tance does the force act? How many ergs of work are done? How 
many ergs of kinetic energy are stored in the moving mass? 


5. A base ball, mass 1^0 gm, was thrown vertically upward and is 
caught by the thrower at the er i of 5.8 sec. Find the height to which 
it rose, the velocity with which t, was thrown, its weight in dynes, the 
work done on it, and the energy stored in it. If the force of the thrower 
was applied during 0.05 sec, what was its amount, exclusive of that re- 
quired to overcome the weight? 

6. A block of marble, 1.00 X 0.50 X 3.00 m, has a mean density of 
2.70 Q^. What is its mass? Express its weight in kilograms, and in 
dynes. Express in ergs and in kilogram-meters the amount of work 
that would be needed to lift it 5 m from the ground. 

7. A brass cylinder has a mass of 122.50 gm, a diameter of 1.90 cm, 
and a length of 5.10 cm. What is its density? 

8. What is the volume of a copper ball whose mass is 130.0 gm, and 
whose density is 8.87 gm? 

9. A pound = 453.6 gm. How many dynes does its weight equal? 
Find the weight of a 130 lb. boy in grams and in dynes. 

10. To how many dynes is the weight of a kilogram equal? How 
many ergs equal a kilogram-meter? 

11. The pile driver. Fig. 20, has a mass of iron weighing 3500 lb. 
This mass is raised by a steam hoisting engine to a height of 45 ft. and 
dropped upon the head of a pile. Calculate the work in foot-pounds 
required to raise the mass of iron to position. How much potential 
energy has it when lifted, and how much kinetic energy when it strikes? 
How much work does it do? 

12. In the case of the pile driver, problem 11, calculate the time of 
falling and the final velocity of the iron mass. From the weight and 
velocity, calculate the energy when striking, and compare the result with 
that calculated from the weight and the height. Which method of calcu- 
lation for the energy would you choose if- both weight and height were 
given, as in this problem? Which if the velocity were known, but not 
the height? 

13. How many pounds of water can be pumped per minute from a 
mine 600 ft. deep by a 75 horse-power pump? 

14. It is desired to raise ore from a mine 550 feet deep at the 
rate of 3 tons per minute; what horse-power must the hoisting engine 
be able to develop? How many foot-pounds of work would it do 
per ton? 

15. An automobile weighing with its load 2000 lb., starting from rest, 
requires 22 seconds to attain a speed of 88 — , when it continues at uni- 
form speed. Calculate its kinetic energy. What average horse-power 
was used in putting it into motion? What other work had to be done? 
How was the energy being expended after the speed became constant? 

16. How many pounds of water must go over a fall each second in 


order to furnish 25 horse-power, if the fall is 10 ft. high and all the 
power is to be used? How many cubic feet of water were used each 
second if 1 cu. ft. weighs 62.5 lb.? What must be the cross-sectional 
area of the stream at the fall, if the speed of the water there is 3 — ? 


1. Consult the libraries on the life of Sir Isaac Newton, and prepare 
a brief paper containing the facts that most interest you. This paper 
may be read before the Physics class, published in the school maga- 
zine, or offered as a theme in the English class. 

2. Repeat Galileo's experiment, by throwing a block of wood and a 
brick from a third story window. 

3. How high can you throw a base ball? Take the time with a 
stop watch, or with an ordinary watch, as accurately as you can; and 
use equation (3). Plot a graph for the complete motion of the ball, 
working out the distances and times for each of the seconds by equations 
(2) and (3). Let the best throwers plot on the blackboard, to the same 
scale, the graphs for their throws. Let the class compare and interpret 
the changes of slope. 

4. Get the necessary data by trial; and calculate your horse-power 
(a) when going upstairs as fast as you can comfortably without a load; 
(h) when carrying the greatest load that you can. 

5. Devise a method of measuring on the wall of the house the greatest 
height to which your lawn hose can throw water. Observe with a 
watch the number of seconds taken by it to fill a gallon jar. Allowing 
8 lb. to the gallon, calculate the number of pounds of water thrown 
out in one sec. From this and the height, calculate the horse-power 
that this stream of water could be made to furnish to a small water 


44. Up Grade. In Chapter II we have learned that a train 
moves because the driving wheels of the engine are turned; and we 
have studied the motions when the track is straight and level. 
There still remain, however, many questions that need considera- 
tion. Why must the engine work harder in ascending a grade? 
How can we find the amount of this extra work? What is the 
relation between the extra pull of the engine and the weight of the 

In all our previous study we have considered motions along 
a straight line, i.e., in one direction or dimension only. The 
questions just asked lead us to the consideration of what takes 
place when a body has at the same time two or more different 
motions. Since these motions may or may not be in the same 
direction, the resultant motion may take place in two or three 
dimensions, i.e., the path of the motion may be a plane curve, or 
it may be twisted like the thread of a screw. 

45. The Composition of Motions. One of the simplest cases 
of two simultaneous motions of the same body is that of a man 
walking lengthwise in a car that is moving uniformly on a straight, 
level track. If the car is moving northward at the rate of 600 ^> 
and the man walks in the same direction at the rate of 150 ^» 
how far does the man travel northward in one second? In three 
seconds? If the man faces about and walks southward in the 
moving car at the rate of 150 ^, how far northward will he travel 
in one second? In ten seconds? 

From these examples it must be evident that in considering 
motions we must take account of two characteristics of the motion, 
namely, direction and magnitude. For this reason it is very 
convenient to represent a motion by a straight line whose length 




and direction correspond to the direction and magnitude of 
the motion. Thus, for the first case just considered, let ab, Fig. 
22, represent the motion of the car northward: cd, which has the 
same direction and is one-fourth as long, will then represent the 
motion of the man with reference to a point in the car; and ad, 
which is obtained by adding together ah and cdy 
will represent the resultant motion of the man in 
both direction and magnitude. 7 

Similarly, in the second 
case, if ab, Fig. 23, represent 
the motion of the train, then 
efy which has the opposite di- 
rection and is one-fourth as 
long, will represent the motion 
of the man with reference to 
a point in the car. Hence a/, 
which is obtained by adding 
together the two oppositely di- 
rected lines, will represent the resultant motion of the man in 
both direction and magnitude. 

It is to be noted that in both cases this addition of the lines 
is performed by drawing the first line with its proper direction and 
magnitude, and then from the end of the first line drawing the second 
with lis proper direction and magnitude. Then the line drawn 
from the beginning of the first line to the end of the second represents 
the resultant motion. 

Fig. 22. Vectors 

Fig. 23 

46. Vectors. In order to indicate clearly the direction that 
such a line represents, it is usually tipped with an arrow point as 
in the figures. A line may be used in this manner not only to rep- 
resent motions, but also to represent any sort of physical quantity 
that has both direction and magnitude. A line that is used to 
represent both the direction and magnitude of a physical quantity 
is called a vector. 

47. The Motions are at Bight Angles. Suppose now that 
instead of walking northward or southward in the moving car, 



the man walks eastward across it. If the velocity of the car 
is 600 ^ northward, and that of the man 150 ^ eastward, what 
is the resultant motion during two seconds? Simple arithmetic 
can not give us a solution that will determine the resultant both 
in direction and magnitude. Therefore let us see what the graph- 
ical method will do for us. 

• In Fig. 24 let the distances northward be represented by the 
ordinates, and the distances eastward by the abscissas, the scale 
being 1 cm = 200 cm for each motion. Plot- y 
ting the graph in accordance with the method 
learned in Chapter I, we find that pi and 'p^ rep- 
resent the positions of the man at the ends of 
the first and second seconds respectively. Will 
the points that represent his position at the end 
of 0.5 sec and 1.5 sec also lie on the line O'p^, 
Will this line include the points corresponding 
to his position -at the end of 0.1, 0.3, 1.9 sec, etc.? 
If we further subdivide the time unit and locate 
points corresponding to any of the hundredths 
of a second, will these lie on the line Opz? Is it 
necessary to subdivide the time unit further in 
order to show that the line O'p^ represents the 
path of the man's motion as accurately as is 
possible in the drawing? 

Fig. 24 gives us the clew to an easy method of 
finding the resultant of any two uniform motions; 
for it is clear that the resultant is represented by 
the concurrent diagonal Opj o^ the parallelogram Oy^ Ji^ ^2> whose 
adjacent sides 0x2 ^^d Oz/j represent the two component motions 
in both their directions and their magnitudes. 

Fig. 24 

Parallelogram op 


48. The Motions are not at BigHt Angles. Furthermore, 
a little careful thought will make clear the fact that whatever may 
he the angle between the component motions, the resultant is com- 
pletely represented by the concurrent diagonal of the parallelogram 
whose adjacent sides represent the two component motions both in 
direction and magnitude. Thus, in Fig. 25, Ox represents one 



of two uniform motions, Oy the other, and the diagonal Op 
the resultant. This construction is called the parallelogram 


49. A Shorter Method. It may already have occurred to 
the reader that the process of finding the resultant may be very 
(/ _-yO much abbreviated. For it is evi- 
dent that we can determine the 
resultant Op2 (Fig- 24) just as defi- 
nitely by means of the triangle OX2P2 
as by the whole parallelogram. In 
order to do this we have only to draw 
the vector 0x2, representing the first 
motion, and from its end Xj to draw 
the vector a^jPa representing the sec- 
ond motion; and then the line Op2> 
which joins the beginning of the first vector with the end of the 
second, is the vector that represents the resultant. 

This method of construction is called the vector method. Fig. 
26 is the diagram for a problem similar to that just considered. 
Since the vector method is simpler than the parallelo- 
gram method and is employed by physicists and engi- 
neers, it will be used in the discussions that follow. 
When we have found the vector that represents the 
resultant, the actual magnitude of the resultant can 
readily be found either from the diagram or by the 
analytical method. --Thus, in Fig. 24 we can measure 
the resultant vector Opj and we find its length to be 
6.15 cm (nearly), and since in this case 1 cm of the 
vector represents 200 cm traversed, the resultant dis- 
tance is 6.15 X 200 = 1230 cm, the result by construc- 
tion and measurement 

Fig. 26 

60. The Analytical Solution. To obtain the analyt- 
ical solution we note that, since the two component motions are at 
right angles to each other, the resultant is the hypothenuse of a right 
triangle; and therefore, since the square of the hypothenuse is equal 


to the sum of the squares of the other two sides, the square of the re- 
sultant is equal to the sum of the squares of the two components. 
Hence, iii the example represented in Fig. 24, since one component is 
1200 cm, and the other, at right angles to it, is 300 cm, the result- 
ant = ^1200H 300* == 1236 cm, the result by the analytical 
method. In general, i: R represent the resultant and A and B 
the two components, then R = ^/Al^ + B\ provided that the com- 
ponents are at right angles with each other. 

Since the two numerical results just obtained represent the 
same distance,, why are they not identical? Would the agreement 
be closer if the diagram were constructed more carefully and on 
a larger scale? 

51. When the Angle between the Components is Oblique. 
In this case the magnitude of the resultant can be obtained graph- 
ically by the addition of vectors in the way just described. Having 
drawn the resultant vector, we measure it in centimeters, and multi- 
ply its length by the number of units that 1 cm represents on the 
scale used in the diagram. 

When the vector triangle is oblique, a purely analytical solution 
is impossible without the use of the elements of trigonometry. 
With a very little knowledge of trigonometry the solution is simple, 
but those who have not this knowledge can always find the result- 
ant by construction and measurement. In fact, it is generally 
more convenient to get the resultant in this way; so that this 
method of solution is very generally used by engineers. 

52. Traveling Crane. The composition of three motions is 
illustrated by a device used in shops where heavy castings or other 
weights have to be lifted and carried from one position in the shop 
to any other. This device, Plate II, is called a traveling crane 
and consists of a steel bridge whose ends rest on little motor cars 
which run on tracks supported by the side walls of the shop, 
so that the crane can traverse the shop from one end to the other. 
The bridge also carries another track along which another motor 
car can run across the shop from one side to the other, while the 
weight to be carried may be lifted to any desired height by means 


of a pulley hanging from the bottom of this car. The motor 
cars and pulley are operated by electricity, steam, or compressed 
air, and the operator controls them by levers sp that the car carry- 
ing the pulley may be made to move either across or along the 
shop while the weight is being raised or lowered by means of the 
pulley. Thus the weight may move vertically while the car carries 
it horizontally across the shop, or the crane may also at the same 
time carry the weight horizontally along the shop. Therefore, 
with this device it is possible to combine motions in three direc- 
tions at right angles to each other. 

53. Besolution of Motions. We have just seen how two com- 
ponent motions may combine to make a single resultant motion. 
In obtaining the solutions of engineering problems it is often con- 
venient to conceive that an observed motion is the resultant of 
two other motions that may have combined to produce it. Thus, 
when by means of the traveling crane a casting is made to move 
diagonally across the shop, it is clear that its actual motion is the 
resultant of two> motions, one across and the other along the length 
of the shop. In a similar way the motion of a railroad train up 
a grade may be conceived as the resultant of two component motions, 
one horizontal and the other vertical. 

This separation of the actual motion into two conceived motions 
leads us at once to the solution of several interesting problems 
connected with the motion of bodies "up hill''; for let us suppose 
that a train, running with uniform speed, is just beginning to 
ascend a grade. How much more work must the engine do in 
pulling the train up the grade than in pulling it for the same 
distance and at the same speed along the level track? It is evident 
that the amount of this extra work depends only on the steepness 
of the grade. Suppose that the grade is 1 : 10, i.e., for every 100 cm 
measured along the track, the track rises 10 cm. In order to 
calculate the amount of extra work. done in pulling the train up 
the grade, we shall, as stated, conceive the motion along the track 
as the resultant of two component motions, one horizontal and 
the other vertical. In Fig. 27, ac is the vector representing the 
motion of the train while passing over 100 cm of track, and ab 

Fig. 27. Resolution of Motions 


and be are the vectors representing respectively the horizontal 
and vertical motions of which we conceive ac to be the resultant. 
Now, since ab is horizontal, no extra work is done by the engine 
in imparting to the train 
the motion represented 
by that vector. But be 
is vertical, and it is clear 
that the engine can not 
impart a vertical motion to the train without doing the work of lifting 
the weight of the train. Hence the extra work done by the engine 
in pulling the train up grade is the work done in imparting to the 
train the motion represented by the vector be. But since the grade 
is 1 : 10, the work done when the train traverses 100 cm of track 
is that of lifting the train through a vertical height of 10 cm. Thus, 
if the mass of the train is 2 X 10^ gna, then, since the acceleration 
of gravity is 980 ^, we find by equation (4) that the weight of 
the train is / = ma = 2 X 10* X 980 = 196 X 10* dynes. Since 
the vertical displacement is Z = 10 cm, we have from equation 
(5) for the work done JF = /Z = 196 X 10* X 10 = 196 X 10^' 
ergs. This, then, is the extra work done on the train by the engine 
for every 100 cn;i up grade along the track. 

54. The Engine is Stalled. If the engine is not able to 
supply the extra energy necessary to do this amount of work, it 
will be stalled on the grade. Let us suppose that this has just 
happened. What ten4ency to motion down grade has neutralized 
that due to the engine pulling up grade? How does the magnitude 
of this tendency depend upon the steepness of the grade? 

55. Force Vectors. In these questions we are dealing with 
forces, not motions; but forces have both direction and magnitude; 
and therefore they can be represented by vectors, provided they 
act at the same point. Thus, in Fig. 28 let the vector Om rep- 
resent the weight of the train, which acts vertically downward. 
This weight produces both a pressure against the track, and a tend- 
ency to move downward along the incline. Hence to answer our 
questions, we conceive the vector Om to be resolved into two 
components, one perpendicular to the track and the other parallel 


to it. The first vector Op will then represent the pressure against 
the track, in both direction and magnitude, and in like manner the 
vector jmi will represent the tendency to move down the incline. 
Now, since the pressure of the train on the track produces no 

motion in the direction 
of the vector Op, it must 
be evident that this pres- 
sure is balanced by an 
equal and opposite pres- 
sure. This equal oppos- 
ing pressure will at once 
be recognized as the reac- 
tion of the track and earth. But if the component represented by the 
vector pm were not balanced by an opposing force, the train would 
move down the incline, i.e., in the direction of pm. At the instant 
when the train is stalled, it is not moving either upward or down- 
ward along the incline, and therefore what must be the direction 
and magnitude of the opposing • force that prevents the down- 
ward motion? How should the vector representing this oppos- 
ing force be drawn in the figure? 

Fig. 28. 

Resolution op Forces on an 
Inclined Plane 

66. Balanced Forces. When two or more forces act simul- 
taneously on a body in such a way that no motion results, these 
forces are said to be in equilibrium. When forces 
are in equilibrium the vectors that represent them 
in the vector diagram, when added together, form 
a CLOSED FIGURE. Thus, in the example just dis- 
cussed there are three forces acting, namely, the 
weight of the train, represented by Ow (Fig. 28), the 
resistance of the track, represented by pO, and the 
pull of the engine, represented by mj). If these three 
vectors be added by laying off one from the end of an- 
other, each with its proper direction and magnitude, and 
in any order, they form a closed triangle as in Fig. 29. forces in 

On the other hand, and in general, if we have any ' quilibrium 
number of forces not in equilibrium acting on a body, and wish 



we add successively the vectors that represent the given forces, 
and draw a line from the end of the last vector to the beginning of the 
first. This line will then be the vector that repre- 
sents the force sought, in both direction and magni- 
tude. Thus, in our example, if we know the weight 
of the train and the resistance of the track, and if 
we wish to find by this method the pull of the engine 
which will hold the train stationary on the grade, 
we add together the vectors rs and st which repre- 
sent the two known forces as in Fig. 30; then 
the line tr is the vector sought. 

This method of finding the force that is able 
to hold a system of other forces in equilibrium is 
very useful in engineering practice in connection with the design of 
bridges, roof trusses, and other structural work in which it is nec- 
essary to determine how strong a beam or tie must be in order to 
resist the given stresses and hold them in equilibrium. 

57. The Pull of the Engine and the tendency down the 
incline are in equilibrium; therefore we can determine the magni- 
tude of either of them, either graphically or analytically. Thus, 
since the mass of the train is 2 X 10* gm, the vector Om (Fig. 28), 
2 cm long, represents the weight, namely, 196 X 10' dynes; and 
hence 1 cm in the diagram represents 98 X 10' dynes. The length 
of the vector pm is found by measurement to be 0.2 cm, and 
hence it represents a force of 0.2 X 98 X 10' = 196 X 10* dynes, 
which is the magnitude sought. 

68. To Get the Analytical Solution we must notice that the 
triangles .450 and Omp (Fig. 28) are similar. (Why?) Therefore 

^^AC' (^^^^ • ) ^"* ^^^^^ *^^ ^^^^^ ^^ AC ^ To ^^'^yP^^^" 

esis, it follows that ^Ic— = — . (Why?) Whence pm = ^V Om.. 

Since Om = 196 X 10\ pm = yV X 196 X 10' = 196 X 10' dynes, 
as in the preceding paragraph. 


69. Less Force: Greater Distance. Now, we have learned 
in Art. 53 that the extra work done in pulling the train 100 cm 
along the incline is the work done in lifting the train through a 
vertical height of 10 cm, i.e., in the case there considered, it is 
W = Jl= 196 X 10* X 10 = 196 X 10'" ergs. But we have just seen 
that the pull' of the engine is 196 X 10* dynes; and therefore, when 
this pull is exerted through a distance of 100 cm, the work done is 
W = /7'= 196 X 10* X 100 = 196 X 10'* ergs, as it should be. 
It will be noted, however, that although the amount of work is 
the same as that previously calculated from the vertical lifting of 
the train, the force of the engine is only -^^ of that which would be 
required to lift the train vertically through the 10 cm. The advan- 
tage of using AN INCLINED PLANE is therefore apparent, since we 
see that by means of it we can do a given amount of work with a 
smaller force than would be required without it. Hence such an 
inclined plane is said to furnish a mechanical advantage. 
This mechanical advantage is defined as the ratio of the resistance 
overcome to the effort applied. In the case of the inclined plane, 
when the effort is applied parallel to the length of the plane, the 
measure of the mechanical advantage has been shown to be the 
ratio of the length to the height. 

Thus, in general, if h represent the height of the plane, I its 
length, R the vertical resistance to be overcome, and / the force 
exerted parallel to the plane (c/. Art. 58 and Fig. 28), then 

? = i 

/ ~ h' 

This is the analytical expression for the mechanical advantage of 
the inclined plane when the effort is applied parallel to its length. It 
may also be written Rh = fl, which expresses analytically the fact 
that the amount of work done by the force applied parallel to the 
plane is the same as that which would be done if the body were lifted 
vertically through a distance equal to the height of the plane. 

It has probably occurred to the reader to ask, Since the engine 

. is stalled part way up the grade because its pull is no greater than 

the pull of the train dow^l grade, why does the train ascend the grade 

at all? The answer is that when the train reached the grade it 



was moving with a uniform velocity; hence it had kinetic energy 
whose amount is determined by equation (6) as e — im F*. It 
was this kinetic. energy that did the work of lifting the train; and 
when this energy was expended, the unaided force of the engine 
could carry the train no farther. 

60, Definitions. Some of the ideas considered in this chapter 
occur so frequently that » we shall do well to frame definitions 
for them. 

The single motion that will produce the same effect as that 
produced by two or more 
motions is called a re- 
sultant MOTION. 

The several motions 
that combine to produce 
the resultant are called 


The process of find- 
ing the resultant of two 
or more motions is called 


The process of find- 
ing the components when the resultant is known is called the 


By substituting the word force wherever the word motion is 
used, we can frame a similar set of definitions for the composition 
and resolution of forces. 

The single force that will hold two or more others in equilib- 
rium is called their equilibrant. The equilibrant of any set of 
forces is equal in magnitude to their resultant, and opposite in 
direction. The point of application of the resultant is identical 
with that of the equilibrant, 

61. The Problem of the Besolution of a Motion, or of a force 
acting at a given point, into two components is indeterminate 
unless something more than the resultant is given. Stated 

Fig. 31. Inclined Railroad, Pike's Peak 


geometrically, the problem is : given one side of a triangle, to find 
the other two. Evidently, we can construct any number of triangles 
that will satisfy this condition. 

A little attention to the geometry of the triangle shows that 
in addition to the direction and magnitude of the resultant, we 
must know of the components either (1) both magnitudes (three 
sides) ; or (2) both directions (a side and two adjacent angles) ; 
or (3) one magnitude and one direction (two sides and an angle). 


1. Any linear motion may be represented in both direction 
and magnitude by a straight line called a vector. 

2. The vector of a resultant motion is found by adding the 
vectors of the component motions. 

3. If two component motions are at right angles to each other, 
the resultant motion is numerically equal to the square root of 
the sum of the squares of the two component motions. 

4. Any motion may be resolved into two or more component 

5. In order to resolve a motion into two components, we must 
know of the components either (1) both directions; or (2) both mag- 
nitudes; or (3) one direction and one magnitude. 

6. The mechanical advantage of an inclined plane is equal 
to the length of the plane divided by its vertical height. 

7. The work done in moving a body up an inclined plane is 
equal to the work done in lifting the same body vertically through 
a distance equal to the height of the plane. 

8. Forces that act at a given point may be represented by 

9. When the vectors that represent any set of forces in equi- 
librium are added together in any order, they form a closed polygon. 

10. The vector that represents the resultant of a number of 
forces not in equilibrium is found by adding in any order the vectors 
of these forces, and drawing a straight line from the beginning of 
the first vector to the end of the last. 

11. The equilibrant of any set of unbalanced forces is equal 
to their resultant in magnitude, but opposite in direction. 



1. Explain what a vector is, and how it may represent completely 
any physical quantity that has direction and magnitude. 

2. Explain how vectors may be added in order to find the resultant 
of two motions when these two components have: 1, the same direc- 
tion; 2, opposite directions; 3, directions that are neither the same nor 
opposite. How is the magnitude of the resultant motion found after 
the resultant vector has been drawn? 

3. Explain the manner in which the analytical expression for the 
resultant of two motions may be found when the components have 
directions at right angles to each other. 

4. Describe a traveling crane, and explain how with it a body 
may be given two or three different motions at the same time. 

5. With the aid of a vector diagram, explain how the motion of a 
body up or down an inclined plane may be conceived as made up of two 
components, one horizontal and the other vertical. 

6. When the weight of a body and the vertical height of an in- 
clined plane along which it is to be lifted are known, what is the amount 
of work done in lifting it along the plane? 

7. How does it follow from the vector diagram in question 5 that 
the work done in lifting the body up the incline is equal numerically 
to the weight of the body multiplied by the vertical distance through 
which it is lifted? 

8. Show by a vector diagram that the weight of a body is to the 
force necessary to hold it in equilibriima on an inclined plane as the 
length of the plane is to the height. 

9. Write an equation which expresses this relation. How does 
this equation show that the inclined plane furnishes a mechanical ad- 

10. By means of this equation, show that the work done by a force 
pushing the body upward along the incline is equal to the work that 
would be done if the body were lifted vertically through a distance 
equal to the height of the plane. 

11. Show how the vector for the resultant of any set of forces 
acting at a point may be found. 


1. A man rows a boat with a velocity of 200 ^"^ southward in a 

"^ sec 

stream that has a velocity of 100 — southward. Find the resultant 

velocity of the boat by the vector method, and also by calculation. 

2. Find the resultant velocity by both methods when the boat is 
rowed northward, the speeds remaining the same. 


3. Find the resultant velocity by both methods when the man keeps 
the boat headed due westward, and does not try to resist the current, 
but rows with the same speed as before. 

4. A boy rows a boat with a velocity of 3 ^^^^, keeping it headed 
across the stream, and not attempting to resist the current. The ve- 
locity of the current is 4 E^. Find the resultant velocity of the boat 
by both methods. 

5. Suppose that the width of the stream in problem 4 is } 
mile, how many minutes will it take to cross the stream? How far will 
the boat drift down stream? How far will it actually travel along the 
resultant path? 

6. The boy wishes to cross the stream in the same time as in prob- 
lem 5, but intends to land at a point directly opposite the starting 
point. Show by vectors the direction in which he must keep the boat 
headed. By both methods find the speed with which he must row in 
order that the boat may move in a straight line from the starting 
point to the landing point. How far up stream would his row have 
taken him if there were no current? 

7. If the traveling crane, Plate II, carries the pair of wheels across 
the shop at the rate of 1.2 — , while it moves along the shop at the 

rate of 1.6 — » ^^ ^^® resultant velocity by both methods, 

8. Suppose that in addition to the other two motions of problem 7 

the crane pulley rises vertically at the rate of 0.5 — , what is the final 
resultant velocity of the pair of engine wheels? 

9. A trolley car weighs 10 tons and moves 1000 ft. along a grade 
that rises 1 ft. in every 100; how much work must the motor do? What 
is the mechanical advantage of the plane? What is the amount of the 
force that moves the car up the grade? 

10. If in problem 9, the speed was 50 — , what was the horse-power? 

11. The height of an inclined plane is 2 m and its length 10 m; the 
weight of a barrel that is rolled up this plane is 150 Kg; required the 
mechanical advantage of the plane, the number of kilograms-force 
exerted, and the number of kilogram-meters of work done. 

12. A ball rolls down a smooth inclined plane whose length is 10^ 

cm and whose height is 10* cm. If the ball had fallen vertically, its 

acceleration would have been 980 -^. Conceive this acceleration to 


be made up of two components, one along the plane, and the other 
perpendicular to it. Determine both graphically and by calculation 
the acceleration of the ball down the incline. 

13. The weight of a kite is 2X10^ dynes; the pull on the string is 
4X10^ dynes and makes an angle of 60° with the vertical. Find the 
resultant pull on the kite. What must be the direction and magnitude 
of the force that keeps the kite in equilibrium? 



14. In Fig. 32, ab represents the direction of the keel of a boat, 
and the line d the direction of the sail, and / is a vector that repre- 
sents the effective pressure of the wind, 200 kil- 
P ograms-force. By the vector _ 

method, find the force that 
urges the boat forward, and also 
that which urges it sideways. 
How is sideways motion pre- 

15. By the vector method, 
find the nimiber of kilograms- 
force with which the beam or 
strut ab, Fig. 33, must push, 
and that with which the tie rod cd must pull in order to keep the 40 
Kg ball in equilibrium. 

Fig. 33 

16. A ball is thrown upward with a velocity of 4900 


how many seconds will it rise before its velocity is reduced to zero by 
the negative acceleration of 980 ^j? What is the distance to which 
it rises in this time? How long will it take to reach the starting point? 
Calculate the velocity at the instant of reaching the starting point and 
compare this with the velocity with which it is thrown. 

17. Calculate the distances traversed by the ball of problem 12 
at the ends of the successive seconds, and plot the graph for its motion. 
Describe the changes of slope. What is the slope at the maximum 
distance, or highest point? Compare this graph with the path described 
by a body thrown obliquely upward. 


1. Point your lawn hose at an angle of 45° elevation; note the 

path of the drops of water. Assuming 1000 — as the initial velocity 

of the water, find its vertical and horizontal components. Assume that 

the horizontal velocity is uniform and that the vertical velocity has a 

negative acceleration of 980 ~. Calculate the distances traversed ver- 

tically and horizontally at the end of each fifth of a second. Plot a 
graph with the vertical distances for ordinates, and the horizontal 
distances for abscissas. Is this graph the same sort of curve as the 
actual path of the water? Are you justified in inferring from the com- 
parison that the vertical velocity was uniformly accelerated and the 
horizontal velocity uniform in the case of the water jet? 

2. Bring a toy sail-boat to the class room to illustrate problem 14. 
With the aid of vectors, can you find an explanation of how such a bo»<^ 
can "beat against th^ wind"? 


3. Bring in sketches or photographs which show struts and ties used 
in ways similar to that mentioned in problem 15. You will find them 
on electric light poles, supporting signs, in the frames under cars, in 
roofs, in bridge trusses, in jib cranes, in locomotive cranes, in bicycle 
frames, etc. Try to draw the vector diagrams for each case brought in. 

4. How high can you throw a ball? Note with a watch the total 
time taken by the ball in rising and falling. Also calculate the initial 
velocities (c/. problems 16 and 17). Place on the blackboard the names 
of the best throwers, with velocities and distances attained. 



62. How Botation is Caused. Thus far^ we have consid- 
ered motion of translation only. We are now ready to take up some 
of the conditions under which rotary motion may occur; and the 
railroad train furnishes us with several questions whose answers 
will help us to describe accurately some relations about which we 
already have some general ideas. How is the translatory motion of 
the piston converted into rotary motion of the drivers? And why 
are the drivers of the fast passenger engine made large, while those 
of the freight engine are made small? 

In order to find the answers to these questions, let us consider 
the diagram. Fig. 34. When the connecting rod pushes on the 

Fig. 34 

crank pin at nj* or pulls at n^, it is evident that it can not cause the 
wheel to rotate, but produces only a useless strain on the moving 
parts. When, however, the crank pin is anywhere above or below 
the line nn^, the pull or push of the connecting rod will cause the 
wheel to revolve. Furthermore, common expe;rience tells us that 
the force of the connecting rod is more and more effective as the 
distance from the center of the wheel to that rod increases. There 
is, then, some relation between the effectiveness of a force in pro- 




Fig. 35. The Moments are Balanced 

ducing rotation, and the distance from the axis of rotation to the 
line of direction in which the force acts. How shall we measure 
the effectiveness of a force for producing rotation? 

Let us suppose that the board in Fig. 35 is supported at the 
middle. It will then balance, so that its weight may be left out of 

the problem. Sup- 
pose that a boy, 
whose weight is 20 
kilograms, sits 100 
cm from the axis. 
If a girl is seated 
100 cm from the axis 
on the other side, the 
boy's weight can just 
hold the girFs in 
equilibrium, provid- 
ed her weight is also 
20 kilograms. Now, if the boy's weight is 25 Kg and his distance 
from the axis is 100 cm, he can balance another at 100 cm whose 
weight is 25 Kg; and so on. Thus in general it appears that the 
effectiveness of a force at a constant distance from the axis of rota- 
tion, is directly pro- 
portional to the mag- 
nitude of the force. 
Again, suppose 
that a boy's weight 
is 20 Kg, and that 
he is distant 200 cm 
from the axis. He 
can now balance two 
children at 100 cm, 
each having the 
weight of 20 Kg 
(Fig. 36). If the 20 Kg boy is distant 300 cm from the axis, his 
weight will be as effective in turning the board as is a weight of 
60 Kg at 100 cm; and so on. Thus, in general, if the distance from 
the axis varies, while the force remains constant, the effectiveness 

Fig. 36. Moment Equals Force X Arm 



of the force in producing rotation about that axis is directly pro- 
portional to the ARM OF THE FORCE, i.e., to the perpendicular dis- 
tance between the axis and the line of direction of the force. 

63, Moment of Force. The effectiveness of a force in pro- 
ducing rotation about an axis is called the moment of the force 
about that axis. 

Since we have seen that the moment of a force is directly propor- 
tional to the magnitude of the force when the arm is constant, and 
directly proportional to the arm when the force is constant, it is 
clear that the appropriate numerical ineoMire of the moment of a 
force is the product of the force and its arm with respect to the given 

Returning to the case of the locomotive drivers (Fig. 34), we see 
that the measure of the turning effect is the force F^, applied to 
the crank pin, multiplied by the perpendicular distance from 
the center of the wheel to the middle line or axis nn^ of the connect- 
ing rod. 

Since now we know how to calculate the moment of a force, we 
shall be able to consider a few problems that will lead us to the 
statement of some very important principles, and will also enable 
us to answer the questions that were raised concerning the relative 
sizes of driving wheels for passenger and freight engines. 

64. The Lever. Suppose that the man in Fig. 37 is to do 
the work of lifting, Avith the lever, a stone which weighs 100 
Kg. He pushes vertically 
downward at one end with 
a force which we will call /. 
The fulcrum, i.e., the axis 
p (Fig. 38) about which 
the lever turns, is distant 
40 cm from the center of 
the stone and 200 cm from 
the man's hands. The 
moment of / with respect ^^°- '^- ^''^ ^^^« 

to the fulcrum is / X 200, and that of the stone's weight is 100 X 40. 
If the moment of / is just sufficient to keep that of the stone's 



weight in equilibrium, then / X 200 = 100 X 40. Whence, 
finally, / = 20 kilograms-force = 20 X 1000 X 980 = 196 X lO' 

The force that will move the stone must, of course, be somewhat 
greater than this, because some unbalanced force is required to 
produce the acceleration. 

The equation may be written: 

100 200 5 
/ " 40 "^ 1 ' 
which states that the mechanical advantage of this lever is 5 (cf. 
Art. 59). 

65. The Work Done by the Lever is easily calculated. When 
the lev^r is moved. Fig. 38, the point s describes an arc with a radius 

of 40 cm, and moves, say, from s 
to s', while the point h describes 
a similar arc with a radius of 200 
cm, going from A to A'. Suppose 
the vertical distance sm through 
which the stone is lifted is 10 cm. 
The effort, at the same time, acts 
through the vertical distance hn. 
If the stone weighs 100 Kg, or 
98 X 10" dynes, calculate how 
many ergs of work are done in lifting it through 10 cm. 

Now, since the right triangles msp and nhp are similar (Why?), 

— == -TTT = T- Since sm = 10 cm, what is the value of 
5m 40 1 

An? Thus it appears that, although by means of this lever we are 
able to dok the work of lifting a stone with a force that is only one- 
fifth of the weight of the stone, this force must be exerted through a 
distance or displacement five times as great as that through which 
the resistance is moved. 

The work done by / is / multiplied by its displacement, or 
(196 X 10^) X 50 = 98 X 10^ ergs. How does this amount of work, 
done by the man, compare with that done on the stone as previously 

40 jn ./ 

Fig. 38. The Lever Diagram 


A lever is often used in another way. as in Fig. 39, when the 
fulcrum is at one end, and the resistance between, — the effort being 
applied at the other end as before. 
In this case the application of 
the principle is entirely similar; 
but the possible mechanical ad- 
vantage is greater, because the / 
lever arm of the effort is longer, t 
The moment of the effort with 
respect to the fulcrum is now 
/ X 240, and that of the resist- 
ance is 100 X 40 as before, the 

... - ^ . .1 Fio. 39. Another Lever Diagram 

mechanical advantage is there- 
fore found from the equation / X 240 = 100 X 40. Whence 

/ = 100 X oTrj = 16.66 Kg-force. The mechanical advantage 

in this case, therefore, is — , or 6. The geometrical construc- 
tion by which the number of ergs of work are found and proved 
equal is much like the preceding, except that the similar trian- 
gles are differently placed. It is easily seen from the figure that 

-f— — -777 = T> and that the effort X 60 = the resistance X 10. 
sm 40 1 

66. The Lever Principle. In the examples just worked out, 
we have learned four things about the lever. Other problems 
involving levers can be solved in a similar manner. The four 
things that we have learned are: 

1. The lever is in equilibrium when the moment tending to 
turn it in one direction is equal to that tending to turn it in the opposite 

2. The mechanical advantage of a lever is equal to the effort 
arm divided by the resistance arm. 

3. The mechanical advantage of the lever may also be obtained 
by dividing the displacement of the effort by the displacement of the 

4. The work done by the effort is equal to tlie work done on the 


These statements may all be verified by very simple experiments 
in which the forces and distances are measured when various 
kinds of levers are in equilibrium. In such experiments and 
problems, it must be noted that whenever the weight of the lever 
itself is at all comparable in magnitude with the other forces 
involved, it also must enter into the calculation. 

67. Equilibrium of Parallel Forces. Another impoiiant fact 
concerning the lever (Fig. 38) is sufficiently obvious without 
argument. The two downward forces must produce a downward 
pressure on the fulcrum; this downward pressure is their resultant, 
and is equal in magnitude to their sum. Hence it is evident that the 
fulcrum must exert an upward resistance which is the equilibrant 
of this resultant, and which is therefore equal in magnitude to the 
sum of the downward forces. It is also clear from Art. 65 that 
the point of application of this equilibrant divides the line joining 
the points of application of the two forces into segments that are 
inversely proportional to the magnitudes of the forces. There- 
fore when a system of parallel forces in one plane acts on a body, 
the condition that must be fulfilled in order that no translatory 
motion may take place is that the sum of the forces acting in 
one direction be equal to the sum of those acting in the opposite 

Similarly, the condition that must be fulfilled in order that 
no rotation may take place is that there be no resultant nioment, 
i.e., that the sum of the moments tending to turn the system in one 
direction about any point be equal to the sum of the moments 
tending to turn it in the opposite direction about the same point. 

It will easily be understood that these conditions for equilib- 
rium which we have seen apply in the case of three parallel forces, 
must hold for any number of such forces; for clearly if there is no 
unbalanced force, there can be no translation; and if there is no 
unbalanced moment, there can be no rotation. 

68. Illustration by a Problem. If we wish to determine the 
single force that will hold a system of known parallel forces in 
equilibrium, we can do so with the help of these principles. For 

-1Q.QL — »50. 



example, suppose it is required to lift the shaft with its pulleys. 
Fig. 40, by applying a single vertical force in such a way that the 
shaft will remain in a horizontal position as it rises. How great 
a force will be neces- 
sary, and at what 
point must it be ap- 
plied? The indi- .^_ M^_^_ 

cated weights of the 

wheels and the shaft A 

are the known forces, T \ ^o 

and their respective So 

*■ ■ Fig. 40. The Shaft Remains Level 

distances from A^ 

the end of the shaft, are the known arms. We may assume that the 
bar is uniform, sd that its weight acts at its middle point, as shown 
in the figure. The lifting force / and its arm r are to be deter- 
mined. From the dimensions on the diagram we see that the sum 
of the downward forces is 50 + 60 + 30 + 80 = 220. The 
required upward force, therefore, must be equal to this sum, or 
/ = 220 Kg-force. 

Since the condition for no rotation is that the moments, taken 
with respect to any point, be balanced, we may select the left end 
of the shaft as the most convenient point of reference. The sum 
of the moments with respect to this point is, then, evidently 
(50 X 25) + (60 X 200) + (30 X 250) + (80 X 350) = 48750. 
This moment must be counterbalanced by that of the upward force 
of 220 having the unknown arm r. Hence, (220 X r) = 48750. 
Whence r = 221.6 cm, i.e., the force necessary to hold the shaft 
in equilibrium is 220 Kg-force; and it must be applied at a point 
221.6 cm from the left end of the shaft. Of course some addi- 
tional force will be required to produce the acceleration when the 
shaft is moved. 

69. The Equilibrant of Any Number of Parallel Forces may 
be determined in a manner similar to that used in the example 
just given. Since we can form two equations in which all the 
forces appear, we may determine either one force and one arm, as 
in the example, or two forces whose arms are known. 



Fig. 41. A Fast Engine 

70. The Locomotive Drivers. Let us apply the principles of 
the lever to the driving wheels of the locomotive. Let F^ repre- 
sent the pull of the connecting rod nn^ (Fig. 34); and let r^, which 

is the perpendicular dis- 
tance from the center 
of the wheel to nn^, rep- 
resent the arm of this 
force. Also let Fj rep- 
resent the push mFj, 
exerted by the rim of 
the wheel along the 
track; and let rg, the 
radius of the wheel, 
which is the perpendicular distance from its center to mF2, repre- 
sent the arm of the push Fj. Then from the equation for the 

F r . 

mechanical advantage of the lever, -^r = — • This equation shows 

^ t ^ 
that the horizontal push at the rim of • the driver is less than that 
exerted on the crank pin, in the same proportion as the distance 
T^ of the crank pin from the center is less than the radius ra 
of the wheel. 

Now, an engine with large driving wheels can develop greater 
speed than can one with smaller ones, because the circumferences 
of the drivers are large; and the engine will go farther, for each 
stroke of the piston. But in this case our equation shows us that, 
other things being equal, the push that can be exerted on the 
track is proportionately 
less; because rj, the 
radius of the driving 
wheel, is increased in 
the same proportion as 
is the circumference. 

On the other hand, 
an engine which is to 
haul a long and massive freight train, must be able to exert a very 
great horizontal push on the track, and therefore rg must be made 
smaller in proportion to r^. This necessitates smaller driving 

Fig. 42. A Powerful Engine 



wheels, giving less speed. Also, since the aim is to get as much 
power as possible, the engine must not only have large and power- 
ful steam cylinders, but must also be very heavy, so as to exert 
sufficient pressure on the track; otherwise the driving wheels will 
slip, and the engine will not be able to move the train. 

71. Weight and Center of Mass. Some very important 
applications of the principles pertaining to parallel forces are found 
in the action of gravity on bodies. 
For gravity tends to pull each 
particle of a body toward the 
center of the earth; therefore, 
the gravity forces that act on all 
the particles of a body are prac- 
tically parallel, and their result- 
ant is the weight of the body. 

Now, when any body is acted 
on by a system of forces affecting 
all its particles, and all in the 
same direction, there is a point 
so situated that the moments of 
all those forces will be balanced 
about any axis that passes 
through this point. This point, 
which is evidently the point of 
application of the resultant of all the parallel forces acting on the 
particles of the body, is called the center of mass. The center 
of mass of a body is, therefore, the point of application of its 
weight, and hence it is often called the center of gravity. 

72. Equilibrium. // a force acting vertically upward, and equal 
to the weight of a body, he applied so that its line of direction passes 
through the center of mass, the body will he in equilibrium under 
the action of this force and its weight 

Thus suppose that c (Fig. 44) is the center of mass of a body sus- 
pended at a point s, about which it is free to rotate, as is the case 
with swing (Fig. 43). Then if the body has been slightly displaced 

FiQ. 43. The Swing 



from the position wherein c is vertically below s, there is a moment 
which is equal to the product of its weight w and the distance sb, 
and which will return it to that position. In what position will 
such a suspended body be in equilib- 
rium? The vase and the pitcher, Fig. 
45, are in equilibrium; what moment 
tends to return each of them, when it 
is slightly tilted? 

73. The Stability of a Body like the 
vase or the pitcher, which rests on a 
BASE, is measured by the amount of 
work that must be done in overturning 
it. A little consideration will show that 
the amount of this work may be de- 
termined as follows: With o as a cen- 
ter and a radius equal to ac, describe an 
arc. This arc is the path that the cen- 
ter of mass c will describe when the 
body is overturned about the point or 
axis represented by a. When the cen- 
ter of mass c is in the vertical line that 
passes through a, the body will be in unstable equilibrium, and 
the smallest further displacement will overturn it. From c draw 
a horizontal line intersecting oc' at a point 6. Then &(/ is the ver- 
tical distance through which the center of mass must be raised in 
order to overturn the body; and the work done is found by mul- 
tiplying the. weight by this vertical distance. 

Other things being equal, if the base of the body were smaller, 
or if the center of mass were higher, as in the case of the vase, 
what would be the effect on fee', and on the work done in over- 
turning the body? 

Answers to questions like these lead to the general conclusion 
that, other things being eqical, the larger the base of a body, and the 
lower its center of mass, the greater is its stability. 

Fig. 46 represents a sphere of uniform density, whose center 
of figure is therefore its center of mass. Show that it is in equi- 

FiG. 44. 

The Swing Dia' 




Hbrium in any position on a level plane. Show also that when the 
plane upon which it rests is slightly tilted, there is a component 

Fig. 45. Stability is Measured by Work 

of force urging it down the plane, and also a moment of force tend- 
ing to rotate it. 

74. Determination of Center of Mass. The foregoing prin- 
ciples of equilibrium enable us to find the center of mass of a body 
by experiment; for if the body be 
freely suspended from a point near 
one of its extremities, it will come to 
rest in the position wherein the arm of 
its weight becomes zero (cf. Fig. 44). 
This position is evidently that in 
which the center of mass is in the 
vertical line passing through the 
point of support. If this line be 
indicated by a plumb line, and 
marked on the body, we know that it contains the center of 

Fig. 46. The Ball Has No Sta- 



If, now, another point of suspension be selected, and a new 
vertical line marked in the same way, it must be apparent that the 
center of mass, since it is in both these lines, can be nowhere else 
than at their intersection. 

Another way of finding the center of mass of a flat, thin body, 
such as a piece of pasteboard, is to balance it flatwise upon a straight- 
edge, and mark on it the axis upon which it balances. This axis, 
in accordance with the definition, must contain the center of mass. 
Therefore, if another axis about which the moments balance be 
located in the same way, the center of mass 
is the* point in which the two axes intersect, 
for it will be found that every other axis on 
which the body will balance passes through 
this point. 

These experiments are of great con- 
venience in connection with certain engi- 
neering problems; for it is often neces- 
sary to find the center of mass of a part 
of a machine, or of a piece of some struc- 
tural work, in order that it may be de- 
signed so as to be in equilibrium under 
the given conditions. 

For example, there must be placed 
on a locomotive driver (Fig. 47) a coun- 
terpoise having a moment exactly equal to that due to the con- 
necting rod or side rod used in turning the wheel. This is be- 
cause if the moments of all the rapidly rotating parts are not thus 
accurately balanced against each other, the system will wabble and 
produce a wasteful and even destructive strain on its axis of rota- 
tion. In order to place the counterpoise properly, its center of 
mass must be known ; and since it is not a regular body, this deter- 
mination can not easily be made by geometry. The usual practice, 
therefore, is to cut out a pasteboard model to a certain scale, 
and experimentally determine the center of mass of this model. 
The center of mass of the real object is then easily located, for it 
is the point of the real object that corresponds to the center of mass 
of the model. 

Fio. 47. The Driver Has 
A Counterpoise 


The same method is used in order to get the position of the 
center of mass of half of a stone arch, so that the moment due to 
its weight can be calculated. 

75. Mechanical Advantage of a Composite Machine. 
Before leaving the study of the- applications of the lever principle, 
let us consider how we can find the mechanical advantage of a 
contrivance like that in Fig. 48, in which the lever principle and 
that of the inclined plane are used simultaneously. 

In pulling the safe up the inclined plane whose height is 100 cm 
and whose length is 400 cm, the mechanical advantage obtained 

by means of the plane (c/. Art. 59) is t- = -jt^ = 4; i.e., the 

weight of the body that can be moved along the incline is four 
times the pull of the rope. Fur- 
thermore, the effort, which is ap- 
plied to the end of the crank, 
has a greater lever arm than has 
the pull of the rope. The effort 
arm in this case is the length of 
the crank, and the resistance ann 
is the radius of the axle. There- 
fore we obtain by this device a /'«• f- Composite Machine 
. •' /PA ^ windlass and an inclined plane, 
mechanical advantage (c/. Art. 

65), which is equal to the ratio of the length of the crank to the 
radius of the axle. If these two lengths are 50 cm and 10 cm 

/ 50 . 

respectively, then -^ = — = 5, i.e., the pull on the rope is five 

times as great as the effort applied at the crank handle. 

Now, if the man applies to the crank handle a force equal to 
the weight of 40 Kg, it is clear that since the mechanical advantage 
of the windlass is 5, the pull on the rope is equal to 40 X 5 kilograms- 
force. Furthermore, since the mechanical advantage of the 
inclined plane is 4, the weight that can be lifted along the 
incline by this pull of 40 X 5 is, (40 X 5) X 4 = 40 X 20 = 800 
kilograms-weight. Thus we see that the mechanical advantage 

of the combination is ;= — ; — = -77- = 20. This mechanical 

\ effort 40 


advantage of the combination can be quickly obtained by multiply- 
ing together the mechanical advantages of the elementary parts; 
e,g,, 5 X 4 = 20. 

Similar reasoning applied to other problems shows that this 
method of procedure will give the mechanical advantage of any 
composite machine, no matter how complicated it may be. Hence 
in general we can find the mechanical advantage of any composite 
rrmchine by multiplying together the mechanical advantages of the 
several elementary machines of which it is composed. 

76. The Law of Machines. We have seen that for the 
inclined plane and for the lever, the work done by the effort 
is equal to the work done on the resistance. Let us now 
see if this is true for the combination of these two devices. 
In the case just discussed, the effort was supposed to be 
40 kilograms-force = 40 X 1000 X 980 = 392 X 10^ dynes. The 
resistance was 800 kilograms-weight = 800 X 1000 X 980 = 784 X 10^ 
dynes. The distance Ij^ through which the effort acts in one 
revolution of the handle is Z^ = 27r X 50 cm; and the vertical 
distance ^ through which the weight is lifted may be found as 
follows : For one turn of the crank, the rope is drawn up a dis- 
tance equal to the circumference of the axle, or 27r X 10 cm. 
Evidently the safe moves the same distance up the incline. But 
since the height of the plane is one-fourth of its length, the 
vertical distance through which the safe is lifted is one-fourth 
of the corresponding distance that it moves along the incline; 

, 27r X 10 cm . ^ _ _ 
I.e., I2 = T = 27r X 2.5 cm. 

The work done by the effort, therefore, is found to .be 
/i^i = (392 X 10^) X (27r X 50) = 392 X 10^ X tt. That done on 
the resistance is f^k = (784 X 1(f) X (27r X 2.5) = 392 X 10^ X tt. 
Thus the two amounts of work are equal. 

Similar reasoning proves that this principle, which we have 
demonstrated in the cases of an inclined plane, of a lever, and of a 
combination of the two, applies to all machines whatsoever. It 
is usually called the law of machines, and is stated as follows: 
The product of the effort and the distance through which it acts is 


equal to the jyroduct of the resistance and the distance through which 
it is overcome; or, the work done by the effort equals that done on the 
resistance. In symbols, 

fA-hh- (7) 

It should be noted that the distances l^ and l^ mu^st always be 
measured in the directions of the corresponding forces. Also, in 
applying this statement to any particular case, it should be bome 
in mind that the total work done invariably includes some useless 
work against such resistances as friction, rigidity of parts, inertia^ 
reaction, and resistance of the air; so that in order to make the state- 
ment precise and perfectly general, this useless work must be under- 
stood to be added in with the useful work. When this has been 
done, it is invariably found that the work done is the exact equiv- 
alent of the energy expended upon the contrivance (c/. Art. 36). 

77. Efficiency. As the cost of the energy used in a manu- 
facturing plant or a system of transportation is a very large part of 
the operating expense, the efficiency of the machinery used is a 
feature of great importance. It often proves to be very poor 
economy to buy machinery of low efficiency simply because it is 

Since some useless work is always done, no machine has an 
efficiency of 100%. No machine can create energy (c/. Art. 36); 
it can only transfer or transform energy that is supplied to it 
from some external source. It enables the user to apply his energy 
in more convenient ways than would be possible without it; but 
the user is always taxed, as it were, a certain per cent of the energy 
for the convenience thus obtained. 

78. Mechanical Advantage from Law of Machines. The 
law of machines enables us to find the mechanical advantage of 
any machine. For we may write equation (7) (Art. 76) in the form 

T — ^y i-^-> *^^ mechanical advantage of any machine is 

equal to the ratio of the displacement of the effort to that of 
the resistance. It is often more convenient to find the mechanical 



advantage of a composite machine by measuring these distances 
than it is to calculate it by multiplying together the mechanical 
advantages of the parts. This is the case with the screw. 

79. The Screw. The thread of the screw is an inclined 
plane wrapped around a cylinder. Fig. 49 shows how the screw 
would look if one turn of the thread were 

In order to turn the screw about its 
axis, a force /^ is applied at the end of the 
lever, or at the circumference of the head; 
and its displacement Z^, for one turn, is 
the circumference described by the point 
of application of this force. 

When the screw is. rotated, either the 
screw itself or the nut in which it turns, 
moves in a direction parallel to the axis. 
Thus when the jack screw (Fig. 50) is turned once around, the 
stone, or whatever rests on the head of the 
screw, is lifted through the distance /j be- 
tween two adjacent turns of the thread. 
This distance, measured parallel to the 
axis, is called the pitch of the screw. Finally 
if /a represent the resistance to be overcome. 

Fig. 49. Screw Thread 


have from equation (7), t ^ ^f which 
/I h 

tells us that the mechanical advantage 
OF THE SCREW is nunierically equal to the 
circumference through v* which the effort is 
applied, divided by the pitch of the screw. 
Since the lever or head of the screw may 
b2 made very large, and the pitch very small, 
this equation shows that the mechanical ad- 
vantage may be enormous, and is limited 
only by the strength of the materials used 
in the construction of the screws. Thus, wagons, locomotives, 
and even large buildings are lifted by means of jack screws. 

Fig. 50. Jack Screw 



Fig. 51 shows how a large house was lifted up a hill 100 ft. high 
with the help of such screws. They may be seen in the picture 
between the timbers and the house. 

80. The Equal Arm Balance, which is generally used for 
comparing masses, is another important application of the law of 
moments. If we wish to weigh a certain quantity of some sub- 
stance, for example a pound of sugar, the mass, of the sugar in one 

FiQ. 61. Jack Screws in Action 

pan is assumed to be one pound when its weight balances a standard 
pound weight on the other pan. For if the balance comes to 
rest with the pointer at zero the opposing moments are equal. 

Hence if /j represent the weight of the sugar, and /^ that of the 
standard pound mass, and if r^ and r^ represent the corresponding 
arms, then the equation for the balanced moments is ^rj = f^r^. 
But r^ = r^, therefore /j = /i, i.e., the weights are equal. This will 
be true if the arms are exactly equ^l, and if the balance comes to 
rest with the pointer at zero under no load. 

And since it was shown in Art. 31 that the weights are piopor- 
tional to the corresponding masses, it follows that if the weights are 
equal the masses are also equal. Thus the law of moments shows 
us that we are correct in our habitual assumption that we can com- 
pare masses correctly by the process of weighing. Of course the 


accuracy of the comparison is dependent on the accuracy of the 
balance and of the masses in the set employed as standards. 

8L Looking Backward. We have now arrived at a place 
in our studies in Physics where it will be well to pause and 
review what we have learned. The principles are really very 
few^ although, as we have begun to see, their applications to every- 
day life and to the devices of modem civilization are countless. 

First, we have learned how uniform and uniformly accelerated 
motions may be accurately and concisely described, and especially 
so by the graphical and analytical Inethods. We then endeavored 
to gain clear notions of the relations of mass and acceleration to 
work, energy and activity, or power; and we found that by means 
of concise equations in which these relations are expressed, many 
important practical problems may be solved. 

We then considered the behavior of bodies in motion and 
of bodies in equilibrium when acted on simultaneously by two 
or more forces; and we found that the resultant motions and 
the resultant forces can be represented with great ease and clear- 
ness by means of vectors. Further, we learned that many compli- 
cated machines are made up by combining the principles of two or 
more of the elementary mechanical devices known as the inclined 
plane, the lever, the pulley and the screw; and that for each of 
these devices a simple numerical relation between the effort and 
the resistance can be established. This is done by compound- 
ing or resolving forces or motions with the aid of vectors, or 
by taking the moments of all the fojces with respect to some 
conveniently chosen axis, and forming the equation for their equi- 

In conclusion, we found that whenever energy is expended 
upon any kind of machine for the purpose of doing work, the sum 
of the useful work and the inevitable useless work is the exact 
equivalent of the energy expended — no more, no less. 

82. Our Future Study of the several forms of energy will 
show us that there is nothing in the study of Physics but the accurate 
description of relations that occur when energy is transferred 
from one portion of matter to another,, or changed from one form 


into another fonn. We can gain knowledge of these changes only 
through observation and experiment 

Phenomena are thus learned and grouped into classes. The 
conditions under which they occur and their relations to each 
other are described in concise statements called laws. With 
the aid gf the reasoning powers and the trained imagination, 
HYPOTHESES are framed for the explanation of these laws. The 
hypotheses are then tested by deducing from them relations which 
follow as necessary consequences. Careful experiments are then 
devised and carried out in order to determine whether or not 
the relations thus deduced are verified — that is, whether they 
are true or not. 

When a hypothesis is found competent to explain every known 
fact that must follow as a consequence of it, and is verified by 
every appropriate experiment that is made in order to test it, it 
takes rank as an established theory. 

By deducing from a hypothesis or theory certain consequences, 
and then testing these deductions experimentally, most of the great 
scientific discoveries have been made. 

The method of study here outlined is called the scientific 
METHOD. Since the history of great scientific discoveries, and 
of the inventions which have always followed in their wake, has 
plainly shown that this method is the only one by which such 
advances have been made, the great advantage of a study like 
Physics is manifest. It is only by training the powers of observa- 
tion and reasoning, and by developing the scientific imaginiation 
in as many people as possible, that individuals can be produced 
who shall continue the progress in discovery and invention which 
is now going on. For discoverers and inventors must not only 
be bom and educated, but they must be supported materially, and 
encouraged by an intelligent interest on the part of the great body 
of people among whom they live and work. 


1. In order that a body may be made to turn about an axis, 
it must be acted on by a force whose line of direction does not 
pass through the axis. 


2. The effectiveness of a force in producing rotation is its 
moment. Moment of force = force X ann of force. 

3. The mechanical advantage of a lever may be found by 
equating the opposing moments taken with respect to the fulcrum. 

T* • 1 * ru X- effort arm 

It is equal to the ratio, — r-r . 

^ resistance arm 

4. The mechanical advantage of a lever is also equal to the 

^ displacement of effort 
* displacement of resistance' 

5. The resultant of two parallel forces having the same direc- 
tion is equal to their sum; it has the same direction as the two 
forces; its point of application lies on the line joining theirs, and 
divides that line into segments that are inversely as the magnitudes 
of the two forces. 

6. In order that any system of parallel forces may be in equilib- 
rium, the sum of the forces in one direction must be equal to the 
sum of the forces in the opposite direction; also the sum of the 
moments tending to turn the system in one direction about any 
point or axis must be equal to the sum of those tending to turn 
it in the opposite direction about the same point or axis. 

7. The equilibrant of any number of parallel forces may be 
fully determined by means of equations formed in accordance 
with this statement. 

8. The center of mass of a body is the point of application of 
the resultant of any set of forces that act in the same direction 
equally on all its particles. 

9. The center of gravity of a body is the point of application 
of its weight, and is identical with its center of mass. 

10. The stability of a body is measured by the work that must 
be done in overturning it. 

11. The mechanical advantage of a composite machine is equal 
to the product of the mechanical advantages of all its elementary 

12. In the case of every mechanical contrivance, the work 
done by it is equal to the work done upon it, or f^l^^ = /jZj. 


13. The work done by every machine includes some useless 

14. The mechanical advantage of any machine is also equal 

^, ^. displacement of the effort ,.,,., 

to the ratio, t^ r i-"j.^i ^i > both displacements 

displacement of the resistance ^ 

being measured in the directions of their corresponding forces. 

15. The mechanical advantage of the screw is equal to the 
. circumference described by the effort 

' pitch of the screw 

16. The equal arm balance is used for comparing masses by 
means of their weights. 


1. How must force be applied to a body in order to make it rotate 
about a given axis? 

2. What is meant by the moment of a force with respect to a given 
axis? What is its numerical measure? 

3. Show how, by equating the moments about the fulcrum of a 
lever, we can find the equation for its mechanical advantage. 

4. Show by geometry that for a lever the displacements are pro- 
portional to the corresponding arms. What, then, is the relation 
between the displacements and the corresponding forces? 

5. Show in the case of the lever that the work done by the effort 
equals that done on the resistance. 

6. What are the conditions that must be satisfied in order that any 
system of parallel forces may be in equilibrium? What kind of motion 
will result if each of these conditions is not satisfied? If neither is 

7. Explain what the center of mass of a body is. 

8. Explain why the weight of a body may be supposed to be a 
single force, acting at its center of mass. 

9. What is the measure of the stability of a body? 

10. Show how the stability of a body may be determined graph- 

11. Show by a diagram that the stability of a suspended body is 
increased by increasing the distance between its center of mass and 
its point of suspension; and vice versa. 

12. Show by diagrams that the stability of a body supported on a 
horizontal plane is increased (a) by increasing the area of the base, 
(6) by lowering the center of mass. 

13. Explain how the principles of stability are applied practically 
m the construction and loading of buildings, wagons, and cars. 


14. Explain why a man on a step ladder is more easily overturned 
the higher he ascends, unless the feet of the ladder are put propor- 
tionately farther apart. 

15. In what two ways may the mechanical advantage of a com- 
posite machine be determined? 

16. State the law of machines, and write the equation that ex- 
presses it analytically. 

17. What kinds of useless work are done by a machine? 

18. Of what commercial importance is the efficiency of machines? 

19. What are some of the uses of the screw? How may its mechan- 
ical advantage be determined? 

20. From the law of moments, show why we can correctly compare 
masses by means of the equal arm balance. 


1. A workman applies 75 Kg-force at one end of a crowbar 200 cm 
long; what weight may be lifted at the other end distant 25 cm from 
the fulcrum? What is the mechanical advantage? With the same 
ratio of the arms, what effort will be required to overcome a resistance 
of 800 lb.? In each case what was the amount and direction of the 
pressure exerted by the fulcrum? 

2. A safety valve lever, Fig. 52, has its fulcrum at a, and is to push 
down on the valve rod at c, which is 2 cm from a. What must be the 

weight of the ball, if it is to be applied at 
notch 5, which is 6 cm from a, and exert 
at c a 5 kilograms-force? What force will 
be exerted at c if the ball weighs 2 Kg and 
is placed at notch 10, which is 12 cm from 

3. Suppose that a trunk 0.9 m long, 0.6 
m high, and weighing 120 Kg is to be tipped 
over on one end by lifting at the other. The weight is uniformly dis- 
tributed: how much force is necessary to start it? Will this force 
increase or diminish as the trunk approaches the upright position? Rep- 
resent graphically the stability of the trunk, and express the value 
of the stability in kilogram-meters of work. 

4. Devise a scheme for weighing a turkey, using only a stick of 
uniform density and cross-section, and 50 cm long, a cm scale, some 
strong cord, and a flatiron known to weigh 2.73 Kg (6 lb.). Be sure 
that your scheme provides for eliminating the weight of the stick. 
Illustrate your method by working out a numerical example. 

5. A bridge whose weight is 3 X 10^ lb. rests on abutments 75 ft. 
apart. Assuming that the weight of the bridge is uniformly distrib- 
uted, what part of this weight is supported by each abutment? What 




additional pressure is applied when an engine that weighs 12 X 10* 
lb. stands with its center of mass 25 ft. from one end of the bridge? 

6. With a single fixed pulley (Fig. 53) , what pull on one end of the 
cord will support a weight of 100 Kg at the other? Use the lever 
principle, assuming the radius of the pulley to be 5 cm. 

In this case what is the pull on the support if the pulley i "^ i 

itself weighs 2 Kg? When the resistance is overcome * — h ■ ' 

through 1 m, through what distance does the force act? V 

What is the mechanical advantage? x- 

7. A movable pulley is arranged as shown in Fig. 54. — ' 

With what force /i, and in what direction, ^ 
J I must you pull in order to support a weight , 

^y^ /j, of 150 Kg? Would the force be the same t 
I if you pulled at /, making use of the fixed 
"^ pulley? Use the lever principle, taking q p 

for one fulcrum and i for the other. Can you V y 
obtain th^ same result from the law of ma- j ^ 

chines [equation (7), Art. 76]? Express the 
mechanical advantage in terms of the lever 
Fig. 53 arms, and also in terms of the distances. ^iq. 54 

8. Show that if the arrangement of pulleys 
in Fig. 54 were turned end for end, the mechanical advantage would 
be 3 for a pull at /. In arrangements of this sort, what is the relation 
between the mechanical advantage and the number. of parts of the cord 
that pull against the resistance? Is there any useless work done by 
the pulleys, so that the mechanical advantage actually obtained is less 
than that given by the calculation? 

9. The screw of a cider press has a pitch of 0.5 cm, and is turned 
by a lever 50 cm long. What is its mechanical advantage? What 
pressure will be exerted on the apples when you apply a 30 Kg-force at 
the end of the lever? 

10. Make a diagram of a combination of any two of the machines 
mentioned in this list of examples. Find the mechanical advantage 
of the combination (c/. Art. 75), and the effort necessary to overcome a 
resistance of 400 Kg-force. 


1. Measure the lever arm and the pitch of the screw of a vise, and 
find its mechanical advantage. 

2. If you are interested in turning lathes, examine one in a shop, 
and see how many of its parts have mechanical advantages. How is 
more force and slower speed obtained by shifting the belt from one 
pair of pulleys to another? When a screw is to be cut, how do you 


make the cutting tool travel along the lathe bed at the desired rate; — 
for example, to cut twice as many threads to the inch as there are 
on the lead screw? What other examples of the composition of mo- 
tions and of the lever principle does the lathe furnish? 

3. What kinds of lever can you find in a sewing machine? in a 
typewriter? in a bicycle? Consult a book on physiology and see if 
you can find the lever principle in the human arm, foot, jaw, etc. 

4. Can you solve the lever problems presented in rowing a boat? 
in using a nut-cracker? in the sugar tongs? in the scissors? in the gas 
tongs? in the claw hammer? 

5. With a set of pulleys like Fig. 54, determine by experiment the 
number of gms- weight at / that will just lift a given weight at W with 
uniform speed. Measure the distance I through which / moves, while W 
is being lifted a distance /i = 10 cm. Calculate / X ^, the work done by 
the effort, and W y^h, the work done on the resistance; also calculate 

the efficiency, . , . Now, by taking off weight at /, find the number 

of gms-wt at / that will just allow W to descend with uniform speed; 
and also find the efficiency in this case as you did in the first. Take the 
average of these two efficiencies as the mean efficiency for the given load. 
In the same way, find the mean efficiencies for, say, 9 other loads, and 
choosing a convenient scale, plot a graph with efficiencies for ordinates, 
and loads for abscissas. Does the efficiency increase with the load? 
In direct proportion, or according to some other law? A set of pulleys 
can be bought cheaply at a hardware store, and will be all the more 
interesting if not too good. How can you determine the number of 
gms-force of the friction? 

6. If mechanically inclined, you may find mines of interesting infor- 
mation about all sorts of mechanical devices, their mechanical advan- 
tages and efficiencies in Perry's Applied Mechanics (Van Nostrand, 
N. Y.), and in Pullen's Mechanics (Longmans, N. Y.). There is much 
in these books that you may not be able to understand; but you can 
read without difficulty enough to increase immensely the knowledge 
that you have thus far acquired. Lodge's Mechanics (Macmillan, 
N. Y.) is easier reading, and will also interest and help you. 

















Note. The authors recommend that this chapter be used for informal 
dbcussion on the first reading. If time is limited it may be omitted. 

83. Flywheels. In the preceding chapter we learned that, in 
order to cause rotary motion, an unbalanced moment of force is re- 

Another important case of the conversion of translatory motion 
into rotary motion is that of a stationary engine and its flywheel, 
Plate III. Here the relations involved in producing the rotary 
motion are the same as in the locomotive and its drivers. But the 
flywheel is large and has a very massive rim, and is designed to 
produce an effect which is not necessary in the locomotive drivers. 

This effect is that of steadying the motion; for it is clear from 
what has preceded that the moment of force acting on the wheel 
is different in different positions of the crank pin, and this will 
cause a jerky motion of the machinery. But the big flyvv^heel 
receives and stores up energy of rotation when the crank pin is in 
the favorable positions, and faithfully pays it back again when the 
crank pin is in the unfavorable positions. Thus it prevents the 
sudden jerks which would be injurious to both the engine and the 
machinery which it runs. 

Why is it that the flywheel has a massive rim and large 
diameter? How does this distribution of the mass make it more 
effective in storing up energy and paying it out again? 

84. Angular Measures. These questions can not be answered 
by expressing the relations in terms of linear velocity, 
because it is clear that different portions of the mass, being at 
different distances from the axis of rotation, have different linear 
velocities Therefore we must have some other means of measuring 
this velocity. Now, it is evident that every spoke of the wheel, and 



in fact every radius, sweeps over the same angle in the 
same time, and therefore all the particles of the wheel have 
the same angular velocity. How, then, is angular velocity 

A convenient way to measure an angle is to find the number 
of times that the radius is contained in the corresponding arc; i.e., 

. _ length of arc 
^ length of radius' 

If in this equation we make length of arc equal to length of 


radius, we have, angle = — = 1. Hence, the appropriate unit 

angle is that angle which corresponds to an arc whose length equals 
that of the radius. This unit angle is called the 
RADIAN, and by a simple calculation is found 
yN. to be equal to 57°.27, Fig. 55. 

/ \ It should be noted that since the numer- 

/ \ ical value of an angle is simply the number of 

f \ times that the radius is contained in the cor- 

'^^ 1 responding arc, it is not expressed in grams, or 

Fio. 56. One Radian centimeters, or seconds; and hence angle has 
no symbol in terms of these fundamental units. 
Now, since the angular velocity is the ratio of the angular space 
described to the time in which it is described, and since the unit 
angle is the radian, angular velocity is measured in radians per 
second; and unit angular velocity is the angular velocity of a body 
which rotates at the rate of one radian per second. Since unit 
angle = 1, the symbol for unit angular velocity is ^. 

If the angular velocity varies, there will be a rate of change of 
angular velocity, i.e., an angular acceleration; and the measure 
of this angular acceleration is, of course, the change in angular 
velocity per second. Since 

, , .. angular velocity 

angular acceleration = — ^ — sL 

^ time 

the unit angular acceleration is one per second, and its 

° sec ^ 

symbol is ^,. 



85. What Corresponds to Mass? We have now learned 
that when we are deaUng with rotation instead of translation, we 
must consider moment of force instead of force, angle instead of 
distance, angular velocity instead of linear velocity, and angular 
acceleration instead of linear acceleration. But what, in the 
former case, corresponds to mass in the latter? 

The answer to this question may be obtained by considering 
the case of a small boy swinging on a gate (Fig. 56). Suppose 
that the gate is open, and that a boy is perched on it at a distance 
of 100 cm from the hinge, or axis of rotation. For simplicity let 
us leave out of consideration the moment of force necessary >to close 

Fig. 56. A Moment of Force Produces Angular Acceleration 

the unloaded gate, and ask how much must be the extra moment 
of force necessary to give the gate a certain angular acceleration 
when there is a second child on the gate with the first (Fig. 57). 
If the masses of the two children are equal, then, since all the condi- 
tions are the same as before except that the mass has been doubled, 
it is evident that the required moment of force must be twice as 
great for two children as for one, three times as great for three 
children as for one; and so on. 

Hence, in general, it appears that when the arm of the mass is 
constant, the moment of force necessary to impart to a mass a given 
angular acceleration is directly proportional to the mass. 

Again, let us ask how the moment of force required to give the 



gate a certain angular acceleration is affected by a change in the 
distance bf the boy from the axis. 

Suppose that the one boy is now 200 cm instead of 100 cm 
from the axis (Fig. 58). The required moment of force will now be 

FiQ. 57. Greater Mass : Greater Moment op Force 

much greater. But how much greater will it be? The mass of 
the boy now has twice the arm, and therefore his moment, with 

Fig. 58. Moment op Force is Proportional to (Arm op Mass)* 

respect to the axis, is doubled, and the moment of force required 
to produce the given angular acceleration would be doubled on 
this account alone. But since the mass must now move through 
an arc twice as long in the same time as before, it follows that 


the moment of force required wouM-'b^'doubled for this reason 
as well. Therefore this moment must Be.2; J< ^ = 2* = 4 times as 
great as before. Similar reasoning shows*'>tl4*^ i^ the boy were 
300 cm from the axis the required moment jis'S^X 3 = 3^ = 9 
times as great as when the boy is 100 cm from ihe,s^&i6p^ In gen- 
eral, then, we find that the moment of force required' ;ta'jtroduce 
the given angular acceleration is proportional not only to the n^iss, 
but also to the square of its distance from the axis. -I^ ..\., 

Furthermore, if we wish to shut the gate more quickly, "a^j 
other conditions remaining the same, a greater moment of force • 
is required to do it. It can be shown by experiment that the 
moment of force must be increased in the same proportion as is the 
angular acceleration that it is to produce. 

Therefore, since the moment of force required to put the mass 
into rotation about an axis is directly proportional to the mass, to 
the square of the distance of the mass from the axis, and to the 
angular acceleration; and since it depends on these quantities 
alone, it follows that we may write as the equation for rotary motion, 
moment of force = mass X (arm of mass)^ X angular acceleration. 

Since for translation 

force = mass X linear acceleration, 
we see that the quantity that stands in the same relation to moment 
of force and angular acceleration as does mass to force and linear 
acceleration is mass X (arm of mass)^. 

This quantity is called the moment of inertia of the mass 
about the given axis and is generally denoted by I. 

86. Rotation vs. Translation. These relations which we have 
.only roughly illustrated in the case of the boys on the gate have 
all been verified by careful experiments and shown to be true in 
all cases; so that in general when we are considering 

ROTATION instead of translation, we must consider 

Moment of force " " Force, 

Angle " " Distance, 

Angular velocity " " Linear velocity, 

Angular acceleration " " Linear acceleration, 

Moment of inertia '* " Mass. 

102 ^ PHYSICS 

Furthermore, we may.obtain the relations that hold in cases of 
rotary motion from^tmlse for the corresponding cases of translatory 
motion simply .bj'.^king the substitutions according to the table 
just given. ^.Eof*^xiample, we have learned (c/. Art. 39) that the 
kinetic eiieygy«6f a body in translatory motion is equal to 

%/;•* mass X (velocity)* 

2 ■ 

c-^Jcence the kinetic energy of a body in rotary motion is equal to 

moment of inertia X (angular velocity)* 

87. Determination of Moment of Inertia. The determi- 
nation of moment of inertia is usually a difficult problem, because 
different particles of the mass are generally at different distances 
from the axis, so that the quantity (arm of mass) is different for 
different particles. Hence, in order to get the value of the 
moment of inertia of any rotating mass, we must first consider 
the moments of inertia of the single particles and then sum 
them all up. 

Thus for the rim of the flywheel. Fig. 59, the moment of inertia 
of a particle on the outside of the rim is the mass of this particle 

multiplied by the square of the outer 
radius of the wheel, and therefore that 
of the layer of particles in the outer rim 
is the mass of that layer multiplied by 
the outer radius squared, because the 
arm of each of these particles is equal 
to this radius. Likewise the moment 
of inertia for the layer of particles in 
the inside of the rim is the total mass 
of the particles in that layer multiplied 

Fig. 59. Moment op Inertia , ., - ., ,. <. .i • 

Equals Mass X (Radius op by the square ot the radms oi the in- 
side of the rim. Now the remainder 
of the mass of the rim may be conceived to be made up of similar 
layers of particles, with radii that are intermediate in length be- 
tween the inner and the outer radius. 


Accordingly the total moment of inertia of the rim is the sum 
of all the products obtained by multiplying the mass of each layer 
by the square of its radius. But since the sum of the masses of 
the layers is the total mass of the rim, and since the radii are all 
intermediate in value between the outer and the inner radius, 
there must be some radius intermediate in value between the 
inner and the outer such that multiplying the total mass 
by the square of this intermediate radius will give the same result 
as would the summing up of all the separate products obtained 
by multiplying the masses of the several layers by the squares of 
their respective radii. This intermediate radius is called the 


The determination of the total mass of the rim is an easy 
geometrical problem, but the calculation for the radius of gyration 
requires the higher mathematics. In cases like that of the rim 
of the flywheel, unless great accuracy is required, we may assume 
that the radius of gyration is equal to half the sum of the inner 
and the outer radii of the rim. 

88. Effectiveness of Flywheels. The effectiveness of the 
flywheel in steadying the motion depends on the magnitude 
of its moment of inertia, and since this magnitude is proportional 
not only to the mass of the wheel, but also to the square of the 
radius of gyration, it becomes apparent that not only must the 
mass be large, but that also this mass must be placed as far as 
practicable from the axis. In fact, we see that a wheel of large 
radius of gyration is just as effective as one of half the radius of 
gyration and four times the mass. 

From this it might seem that we could increase the effectiveness 
of the flywheel indefinitely by increasing its radius of gyration 
without correspondingly increasing its mass. There is, however, 
a limit which can not be passed with safety, because the flywheel 
may burst. A little careful reasoning with the assistance of our 
algebra and geometry will enable us to see why this is true, and 
also to arrive at a very important general principle. 

89. Conditions for Circular Motion. Let the circle p c q, 
whose center is o. Fig. 60, represent a circle on the rim of the fly- 



wheel, and p the position of a small portion m of its mass at a 
given instant. Suppose that the wheel is rotating so that the 
linear speed along the arc is uniform. 

According to the first law of motion, m, if not a part of the 
rigid wheel, would move in the direction of the tangent to 

the circle at p; but since it is con- 
strained to remain on the arc, it 
will at the end of a very short time t 
arrive at some point c on the circum- 
ference. Since t is very small, the 
arc pc will be so nearly equal to the 
chord that we may without appre- 
ciable error regard the arc and chord 
as identical. We may now let pc 
be the vector that represents the mo- 
tion along the arc. As in the case 
of the inclined plane, we may re- 
solve this motion into two components, one in the direction of the 
tangent and the other in that of the radius po. The lines pb and 
he will be the two component vectors. 

Now if we extend the radius po to cut this circumference at 
q, and draw qc, the right triangle, pcq and pbc are similar (Why?); 

Fig. 60. 

The Acceleration to- 
wards THE Center is — 

therefore we have — 

be pc \ , pe^ 

— = ^-— ; whence, be = ^—, 
pe pq pq 

But be repre- 

sents the distance that m, starting at p, traverses in the direction 

af , , . 

po in the small time t; and hence it is equal to — , in which a is 

the linear acceleration of m in that direction. Also pe represents 

the distance that m traverses in the time t with the uniform velocity 

along the arc pe\ so that if v represent this velocity, pe = vt (Why?). 

Further, pq is 2r, i.e., twice the radius of the circle. Therefore, 

when we substitute these values in the foregoing equation, we have 

af v^f 

— - = -^. Solving this for a, the acceleration toward the center. 

we obtain a = — 

The centrally directed force (sometimes called centripetal 



force) that causes this acceleration — must be [cf. equation (4), 

It means that a mass, 

Art. 27'\ f = ma = — . This force, of course, must have an 

equal and opposite reaction, which is often called centrifugal 

The conclusions expressed by this equation will follow for 
any other small time t and for any other point of the circumfer- 
ence: hence the equation 

' ~" r 
applies to all cases of rotary motion. 
m order to move with 
uniform linear speed 
around the circumference 
of a circle, must be acted 
on by a constant force 
whose direction is always 
toward the center, and 
whose magnitude is di- 
rectly proportional to the 
mass and the square of the 
linear speed, and inversely 
proportional to the radius. 
If m is expressed in 

S°^''^^^I^' ^^^ ^^" ^"^' 
then / will evidently be in 

FiQ. 61. 
The Centripetal Force is 

Loop the Loop 


gm y cm2 
cm ^ sec2 

^^^, which will be recognized as the symbol for 


'90. Why Wheels Burst. It is often convenient to express 
this relation in terms of angular units instead of linear units. In 
order to do this we must substitute angular velocity for linear 
velocity and moment of inertia for mass, according to the table, 
Art. 86. Thus, if u represent the angular velocity 

. _ mv^ 
' r 


= mrur. 




This equation shows that if we increase the radius of a fly- 
wheel indefinitely, while the angular velocity u and mass m remain 
the same, the force required to hold the parts together will soon 

become greater than the cohesive force 
of the particles and the wheel will then 

The equation also shows that for a 
given wheel, for example, an emery wheel, 
Fig. 62, the force required to hold the 
parts together increases as the square 
of the angular velocity u\ and hence if 
we continue to increase the number 
of revolutions per second of the wheel, 
a limit will be soon reached beyond 
which the angular velocity can not be in- 
creased with safety. 

Fig. 62. The Emery Wheel 
MAY Burst 

91. Distribution of Mass. There 
is another very important condition 
which must be observed in the construction of a flywheel, or in 
fact of any other rotating part of a machine. The equation 
/ = mrt^ tells us why this condition must be complied with, 
for it is evident that for a given body rotating about an axis, 
all the small masses m have the same angular velocity u) and hence 
the centrally directed force required to keep each such mass moving 
in its circle, depends on the value of the quantity mr corre- 
sponding to this mass. This latter quantity mr or mass X (arm of 
mass) is often called moment of mass, just as force X (arm of 
force) is called moment of force. Now, unless the central force for 
any mass m on one side of the axis be balanced by an equal and 
opposite force on the opposite side of the axis, there will be art un- 
balanced lateral strain on the axis, and the wheel will wabble. 
Hence it is clear that the condition that must be fulfilled to 
prevent the wabbling is that every moment of mass on one side^ 
of the axis must be balanced by an equal moment of mass on the 
opposite side. In other words, the moments of mass must be symmet" 
rically disposed about the axis (cf. Art. 74). 


92. Bailroad Cnrves. Since the relation / = — , which 

we found to hold in the case of the flywheel, applies to every mass 
that is moving with uniform linear speed in a circular path, it must 
apply to the case of a railway train when it rounds a curve. 
Railway curves are usually arcs of circles, and the portions of 
straight track at the two ends of the arc are tangents to the circle 
of which the curve is a part. 

In accordance with the first law of motion, the car, at the instant 
when it reaches the curve, tends to continue moving with uniform 
velocity in the direction of the tangent. For have not all of 
us had the experience of being apparently thrown against the 
side of the car when it began to round a curve unexpectedly? 
The car turns out of its straight path because it is pushed laterally 
by the track; but the passenger continues in the straight path 
until he is suddenly pushed into the new, direction by the side of 
the car. It is clear, then, that to keep the car moving in the curve, 
a horizontal force must continually be exerted by the track; and 
that this horizontal force must act so as always to be perpendicular 
to the track at the point where the pressure is exerted. This 
lateral force exerted by the rails is evidently directed inward toward 
the center of curvature, i.e., along the radius at the point where 
the push is exerted; and, as we have seen in the case of the rim of 

the flywheel, it must be equal in magnitude to , where m is 

the mass of the car, v the uniform speed along the curve, and 
r the radius of curvature of the track. Now, how must the track 

be built that it may exert the central force with the minimum 

strain on both itself and the train? This is accomplished by 
inclining the road-bed so that the outer rail is higher than the 

. Let xyz^ Fig. 63, represent a cross-section of the roadbed 
and c the center of mass of the car, where all the forces may be 
supposed to act. Since the pull of the engine is balanced against 
the friction and air resistance^ these forces may be left out of the 
problem. The forces that must be supplied by the track are, 
first, a force vertically upward and equal to mg, the weight of the 



train; second, a force horizontally inward toward the center of 

curvature and equal to ■ 

These forces are represented by the 

vectors hp and pc respectively; therefore the resultant F of these 

two forces is represented by the. vector 
he. Now, in order to exert this force 
with the least possible strain on the 
rails, the roadbed must be perpendicu- 
lar to he. When this is so, we see from 
the geometry of the diagram that 

FiQ. 63. The Curved Track 
IS Inclined 

zy pc r V" , ^, 
— ^ = f- = ^ = — , I.e., the necessary 
xz op mg rg '' 

lateral slope of the track is directly pro- 
portional to the square of the velocity 
and inversely proportional to the radius 
of curvature. Furthermore, it is not 
affected by the mass of the car, since the 
mass cancels out of the expression. 
The student will recall the similar cases of the inclination of 
a circus ring or race track and the inward leaning of the horse 
and rider; also the impossibility of mak- 
ing a turn with a bicycle on a slippery 
pavement without slackening speed. 

93. Spinning Tops. Another inter- 
esting and important fact about rotary 
motion is that the inertia of a rotating 
body shows itself not only in the tend- 
ency to continue rotating when started, 
but also in the resistance which it offers 
to any force tending to change the direc- 
tion of its axis of rotation. This is well 
illustrated in the case of the top. Fig. 
64, which will stand on its point, and resist any force tending to 
overturn it, only so long as it is rapidly spinning. A similar case 
is that of a rifle ball, which is given a rapid spin about its long- 

FiG. 64. The Top Stands 
ON ITS Point only while 


est axis by cutting helical grooves in the rifle barrel. Since this 
longest axis of the projectile coincides with the path in which it 
was started, the bullet tends to continue in its path point fore- 
most, and to strike in that attitude. 


1. The unit angle is the radian; it hs^s no symbol in terms 
of gm, cm, and sec. 

2. The value of the radian in degrees is 57° .27. 

3. The unit angular velocity is one radian per second. Its 
symbol is ^. 

4. The unit angular acceleration is one radian per second 
per second. Its symbol is ^. 

5. The moment of inertia of a rotating particle is mass X 
(arm of mass)*. 

6. The relations of moment of inertia to rotation are the same 
as those of mass to translation. 

7. The equations of rotary motion may be obtained from 
those of translatory motion by substituting moment of force for 
force, angle for distance, angular velocity for linear velocity, angular 
acceleration for linear acceleration, and moment of inertia for mass. 

8. The moment of inertia of an extended mass is equal to the 
sum of the moments of inertia of its separate particles. 

9. The numerical value of the moment of inertia of an extended 
mass may be obtained by multiplying the total mass by the square 
of the radius of gyration. 

10. A body will not move in a curved path unless it is con- 
stantly acted on by a central force that gives it a uniform accelera- 
tion toward the center of curvature of the curved path. 

11. The numerical value of the necessary acceleration toward 

the center in linear units is — , and in angular units it is rv*. 

12. The force / that will keep a body moving uniformly in a 
circular path is equal to . 

13. Moment of mass is mass X (arm of mass). 

14. In order that a body may rotate smoothly about an axis. 


the moments of mass of all its particles must be symmetrically 
disposed with respect to that axis. 

15. A rotating body resists any force tending to change the 
direction of its axis of rotation. 


1. What is the use of the flywheel of a stationary engine? 

2. What is meant by the angular velocity of a rotating body? Is 
it the same for all particles of the body? 

3. What is meant by the radius of gyration of a rotating body? 
How can the moment of inertia of an extended mass be calculated 
when its radius of gyration is known? 

4. Why is it important in the case of a flywheel to have a large 
radius of gyration? How is the wheel made so as to secure this result? 

5. How may material be economized in the construction of such 
a wheel, and why? 

6. Explain why a flywheel or an emery wheel will burst if of too 
large diameter, or if rotated too rapidly. 

7. With the aid of a vector diagram, show why a curved railway 
track must be inclined inward toward the center of curvature. Give 
some examples of similar cases. 

8. Mention two ways in which the inertia of a rotating body mani- 
fests itself, and illustrate by examples. 


1. Since a circumference = 2ir X radius, 360° = how many radians? 
How many degrees are there in one radian? 

2. An emery wheel makes 2400 revolutions per minute; what is its 

I 1 'J. ' radians^ 
angular velocity m ? 

3. A moment of force whose average numerical value is 4 X lO** 

gives to the flywheel of an automobile an angular acceleration of 
rfl-fiijins •* 

2 5—. What is the moment of inertia of the wheel? How many 

sec^ ,. -^ 

•po li 1 f) Tl Q 

seconds are required to give this wheel an angular velocity of 20 ? 

4. The rim of a flywheel has a thickness of 20 cm, an inner radius 
of 90 cm, and an outer radius of 110 cm. What is its volume? If 
its density is 8, what is its mass? Taking the mean radius as the radius 
of gyration, calculate the moment of inertia of the rim. 

5. How many dynes must act with an arm of 20 cm in order to 

give the wheel of problem 4 an angular acceleration of 1 ? 

6. An emery wheel of 15 cm radius is making 20 revolutions per 


second. How many dynes are required to keep each gram of emery 
at the circumference from flying off? 

7. In a loop-the-loop, Fig. 61, when the car is at the top of the 
loop, what will happen unless the centripetal force necessary to keep 
the car moving in the circular path is equal to or greater than the 
weight of the car? Let m represent the mass of the car, 980 the accel- 
eration of gravity, v its linear velocity at the top of the loop, and r the 
radius of the loop, 'and show that the car will not fall if m X 980 = 

— . Need the mass be considered? If the radius of the loop is 

500 cm, what must be the velocity r? 


1. See if you can swing a small pail full of water around in a ver- 
tical circle without spilling the water. Is this experiment like a loop- 
the-loop? If your arm is 75 cm long, what must be the number of 
revolutions per second when the water does not spill? 

2. Can you find out how the governor of a stationary steam engine 
works? What can you find out from the laundryman about centrif- 
ugal drying machines? 

3. If you have occasion to take the rear wheel off your bicycle, 
hold it by the step, spin it rapidly, rest the end of the step on your fin- 
ger, and see what the wheel will do. 

4. For interesting information about tops, gyroscopes, and rotation, 
consult Hopkins, Exj)erimental Science, (Munn & Co., N. Y), 
pages 10-37. 


94, Pnmps. In the foregoing chapters, we have found it both 
convenient and interesting to learn some of the principles of 
Physics, and some of the methods of investigating physical 
phenomena, by considering the motion of a railway train. We 
found, that in order to answer only a few of the questions which 
naturally arise in connection with the motions of a locomotive 
and its parts, it was necessary to master much of that part 
of Physics which deals with forces and their effects, and is called 
Mechanics. When we take up the study of that form of energy 
called Heat, we shall have frequent occasion to refer to the steam 
engine; and in fact, if we wished thoroughly to understand the work- 
ing of every part of the highest type of modem locomotive, and 
explain all the physical phenomena that occur in connection with 
it, we should find before we had finished that we needed to know 
the greater part of what there is to be known about Sound, Light, and 
Electricity, as well as about Mechanics and Heat. But, as we are 
now to take up the study of the Mechanics of Fluids, we shall find 
more obvious relations in that class of machines called pumps; 
because the specific purpose of every pump is to propel some liquid, 
like water, or some gas, like air or illuminating gas, and to deliver 
it under pressure at places where it is to be utilized. Furthermore, 
we shall gain a much wider view of the value-of such knowledge and 
training as the study of Physics can give us, by finding out some- 
thing of a few other great inventions that contribute largely to our 
modem civilization, and are as closely related to our everyday lives 
as is the locomotive engine. 

In Plate IV is shown one of the powerful pumping engines that 
distribute the water supply of a great city; and Fig. 65 shows a 
very similar machine designed to distribute compressed air for 
operating drilling machines and other appliances used in mines and 


Plate IV. Pump in the 39tii St. Station, Chicago 

This Pump has a capacity of 25,000,000 gallons a day. The pipes In front of the 

picture contain the valves, the plungers for pumping are in the 

cylinders farther in the rear. 


factories. These mammoth pumps grew out of very small be- 
ginnings and were many centuries in coming to their present state 
of power and efficiency. 

95. Lift Pump. The common lift pump was known and used 
in the time of Aristotle. The diagram, Fig. 66, shows how it is con- 
structed and how it acts. It consists of a hollow cylinder at the bottom 
of which is a valve opening upward like a trapdoor and called the 


Figure 65. Air Compressor 

inlet valve. Fitting closely into the cylinder is a piston perforated 
by a hole over which is fitted another valve, also opening upward 
and called the outlet valve. A long pipe called the suction pipe 
extends into the water below. When the piston is lifted by means 
of the piston rod, the outlet valve remains closed, while the inlet 
valve opens, and air from the suction pipe enters the cylinder. 
When the piston is pushed down, the inlet valve closes, so that when 
the piston tends to compress the air in the cylinder, this air, by its 
reaction, opens the outlet valve and passes through the perforation 
in the piston. At the end of the stroke the air that had entered the 



cylinder is above the piston and 
is lifted out by the piston during 
the next stroke The next few 
strokes remove the remainder of 
the air from the suction pipe; and 
the water which follows the air up 
the pipe and into the cylinder, 
passes through the pump and is 
pushed out in precisely the same 
manner as was the air. 

But what force is it tliat 
pushes the air and the water into 
the suction pipe, and causes it to 
lift the inlet valve and flow into 
the cylinder? Aristotle and his 
* followers offered in explanation 
the saying that Nature abhors a 
vacuum; but the reader will rec- 
ognize that this is not an expla- 
nation at all. 

96. Force Pump. During the first century a.d., the force 
pump (Fig. 67) was invented by a philosopher named Ctesibius of 
Alexandria. This differs from the lift pump 
only in that the piston is not perforated; 
and the outlet valve is at the end of the 
cylinder near the inlet valve. The outlet 
valve must of course open outward. Ctesi- 
bius also invented a double acting force 
pump for putting out fires, which differed 
but little from the hand fire engines now 
used in villages. He knew a great deal 
about how pumps worked; but discovered 
nothing that enabled him to explain why 
they worked. The idea that Nature ab- 
horred a vacuum seemed sufficient to satisfy 
the minds of most men until the seven- 
teenth century, when there arose a most fiq. 67. Force Pump 

Fig. 66. The Lift Pump 


remarkable group of men, who in their search for truth about 
the things of the material world began to use the scientific 
method (cf. Art. 82). The result was that they inade more 
discoveries and contributed more to accurate scientific knowl- 
edge in a few years than did all the philosophers in all the 
centuries before them. 

97. Air Has Weight. Galileo had proved that air has weight 
by weighing a glass globe, forcing more air into it, and weighing it 
again. The difference between the two weights, he rightly ascribed 
to the weight of the air that had been added. He did not discover 
that the weight of the air had anything to do with Nature's alleged 
horror of a vacuum. He was astonished when informed that a 
lift pump had been made with a suction pipe about forty feet 
long, and that no amount of pumping would cause the water to rise 
higher than about thirty-three feet. Since a vacuum remained in 
the cylinder and upper part of the suction pipe, he was led to remark 
that the horror of a vacuum was a force that had its limitations and 
could be measured by the column of water that it would raise. 

Galileo's friend and pupil Torricelli (1608-1647), who succeeded 
him as professor at the Academy of Florence, took advantage of 
this suggestion, and began a series of experiments which led him 
to infer that the weight of the water column in the suction pipe was 
supported by the weight of the atmosphere that rested upon the 
surface of the water in the cistern." Reasoning that since mercury 
is 13.6 times as dense as water, the weight of the atmosphere ought 
to be sufficient to balance that of a column of mercury only 
about one-fourteenth as long as the water column, he caused 
two of his pupils to carry out the experiment, which is known by 
his name. 

98. Torricelli's Experiment. A glass tube, about 33 inches 
long, was closed at one end and completely filled with mercury. 
When the open end of the tube had been closed by the finger, and 
the tube inverted, it was supported in a vertical position with the 
open end in a dish of mercury. On removing the finger, the mer- 
cury sank down a little way in the tube, and, after a few oscillations, 


came to equilibrium with the surface of the mercury inside the 
tube about 30 inches (76 cm) above that of the mercury in the dish. 
In the upper end of the tube was a very nearly perfect vacuum. 
Torricelli noticed that the height of the mercury column often 
varied; and he inferred that the variations were due to the changes 
in the pressure of the atmosphere which was "now heavier and 
dense, now lighter and thin." 

Torricelli^s hypothesis as to the pressure of the air was thus con- 
firmed so far as his experiments went, but other experiments 
were necessary in order to establish its truth. 

99. Pascal. When Pascal (1623-1662), who had been studying 
the phenomena of fluids in equilibrium, learned of Torricelli's 
experiments he repeated them, and concluded that "the vacuum is 
not impossible in Nature, and she does not shun it with so great 
horror as many imagine." Pascal reasoned that if one were to 
ascend a mountain, the pressure of the air at the greater elevation 
should be less, because there would be less air overlying the moun- 
tain top than there was overlying an equal area of the plain. Ac- 
cordingly he wrote to his brother-in-law, who lived near the Puy de 
Dome, an ancient volcano in the Auvergne, France, asking him to 
ascend the mountain with a Torricellian tube and observe whether 
the mercury column would not fall because of the diminished 
atmospheric pressure. The experiment was made; and it was found 
that the mercury column became three inches shorter during the 
ascent, but gradually resumed its previous length during the descent 
to the plain. 

Pascal also repeated Torricelli's experiment with wine instead 
of mercury; and he found as he had inferred, that, since wine is less 
dense than water, the atmosphere balanced a column of it which was 
longer than the water column; for of course it would take a longer 
column of the lighter fluid to make the same weight. 

The hypothesis of Torricelli and Pascal as to the pressure of the 
atmosphere was thus placed upon a firm experimental basis, and 
was now competent to explain the phenomena of pumps; but it 
required the evidence of many more experiments to secure its gen- 
eral acceptance. 


100. The Mercurial Barometer. The mercurial barometer 
which is an instrument of great precision, and of inestimable value, 
is simply a Torricellian tube in which the dish for the mercury is re- 
placed by a flexible bag of chamois skin. The tube and bag are 
enclosed in a metal case which is fitted with a very accui^ate scale 
by means of which the height of the mercury colunm may be 

Since changes in the weather are caused by the passsing of areas 
OF LOW PKESSURE, the barometric column falls when one of these 
areas is approaching, and rises again after the low pressure area has 
passed and a high pressure area has taken its place. 

By means of barometers and other instruments, read simul- 
taneously at scores of stations, the U. S. Weather Bureau officials 
are able to map the weather conditions of the entire country 
every eight hours; and thus, as the areas of low or of high pressure 
travel across the country, taking with them their characteristic 
weather conditions, the forecast official announces by telegraph the 
probable time of its arrival and the kind of weather that may be 
expected to accompany it. These weather forecasts and storm 
WARNINGS, which would be impossible without the barometer and 
thermometer, save many lives and much property annually. . 

The barometer is also much used in measuring elevations, 
such as the heights of mountains and the altitudes attained in 
balloon ascensions. Near sea level, the barometer falls 0.1 inch, 
or 2.54 mm, for every 80 feet of elevation; but at greater elevations, 
since the density of the air is much less, the change of elevation corre- 
sponding to a barometric depression of 0.1 inch is greater than 80 
feet, and increases steadily with the increasing elevation. The 
reason why the upper layers of atmosphere are less dense than the 
lower layers is that those upper layers have much less air above 
them pressing down upon them. With a good barometer a differ- 
ence of four feet in altitude can be detected. 

Since the pressure of the atmosphere on 1 cm^ exactly balances 
the weight of a column of mercury having a certain length and 
equal cross-sectional area, we can calculate this pressure in grams 
or dynes per square centimeter by calculating the weight of this 
column. Thus, at sea level, the average height of the bar- 



rometer column is 76 cm, and the density of mercury at 0° Centi- 
grade — ^the freezing point of water — is 13.59 ^. The volume of 
a mercury column 76 cm high and 1 cm* in sectional area is 76 cm'. 
Its mass, therefore, is equal to the product of its volume and its 
density, i.e., M = FD = 76 X 13.59 = 1032.84 gm. 

Its weight in grams is therefore represented by the same number. 
The average pressure of the atmosphere at sea level is thus found to 
be 1032.84 ^^,. 

Let the student substitute 1032.84 for m in equation (4), Art. 
27, and 980 for a, and find the average pressure of the atmosphere 
in dynes per square centimeter. 

101. Characteristics of Fluids. We must now return to the 
researches of Pascal on fluids. Both liquids and gases are classed 
as FLUIDS, because they both have the 
property of offering no permanent resist- 
ance to forces that tend to change their 
shape. Any portion of a fluid acted on by 
forces not equal in all directions flows 
freely in the direction in which it is urged 
by the greater pressure. Furthermore, all 
fluids have perfect elasticity of volume; 
that is, if they are compressed ever so 
much they immediately resume their 

FiQ. 68. Transmission op former volumes when the additional pres- 
Fluid Pressure . , * 

sure IS removed. 
Recognizing these two familiar properties, Pascal reasoned 
about fluids somewhat after this manner: Let the bottle. Fig. 68, 
be completely filled with any fluid, and let the little circles represent 
the elastic particles of the fluid. Suppose the end of the stopper to 
have an area of 1 cm*, and let it be pushed in so as to exert a 
pressure of 1000 dynes. This pressure of 1000 dynes per square 
centimeter will act directly upon the layer of particles adjacent to 
it, so that 6, for example, will be pushed downward against 11 and 
16, and this pressure will be transmitted without loss through 21, 
to the bottom. The pressure transmitted by 6 to 7 and 8 will 
tend to push them to the right and left respectively, so that the 


pressure is transmitted undiminished to the sides of the bottle. 
Furthermore, 8 for example will tend to force 3 upward and 13 down- 
ward, thus transmitting the same pressure to those portions of the 
top and bottom that lie above and below them. Since all the par- 
ticles will be affected in precisely the same manner, it follows that 
the pressure will be transmitted not only in the directions men- 
tioned, but also in every other possible direction; and therefore the 
pressure of 1000 dynes exerted on one square centimeter of the 
fluid will cause a pressure of 1000 dynes on every other square 
centimeter of resisting surface with which the fluid is in contact. 
If the bottle is strong enough to resist this pressure, all parts of the 
fluid will be in equilibrium; for if it were not, any portions of the 
fluid affected by the unbalanced pressures would move freely in the 
directions of the unbalanced pressures until all the pressures were 

The direction of the pressure on any part of the surface must 
be perpendicular to that part; for if the pressure is not perpen- 
dicular to that part of the surface upon which it acts, it may be 
resolved into two components, one perpendicular to the surface and 
the other parallel to it (c/. Art. 53). This parallel component, if it 
were exerted, would tend to rotate the bottle about some axis; and 
since no such tendency has ever been detected, we believe that no 
such parallel component exists. Therefore the pressures are all at 
right angles to the surfaces on which they act. It should be care- 
fully noted that the truth of this reasoning does not depend in any 
way on the shape, size, or number of the particles; and the reader 
should avoid the notion that the particles are spherical, or that the 
diagram in any way represents their size or number. 

102. Pascal's Principle. Having arrived at these conclusions, 
Pascal announced them in the following concise statement, which 
is known by his name : A pressure exerted upon any part of a fluid 
enclosed in a vessel is transmitted undiminished in all directions, and 
a^ts with equal force on all surfaces of eqvxil area, in directions per- 
pendicular to those surfaces. 

A thorough understanding of this principle enables us to ex- 
plain all the phenomena of fluid equilibrium. 




69. The Forces are Pro- 
portional TO THE Areas 

103. Hydraulic Machines. If, for example, we have a vessel 
(Fig. 69) consisting of a large and a small cylinder connected by a 
pipe and each fitted with a water-tight piston, and if the sectional 

areas of these pistons are 1 cm* and 
100 cm* respectively, a force of 1 kil- 
ogram exerted upon the smaller pis- 
ton will produce a pressure of 1 
kilogram per square centimeter on 
the larger piston, or a total force of 
100 kilograms. Thus the force trans- 
mitted to any surface by a fluid is 
directly proportional to the area of 
that surface. "Hence, " said Pascal, 
"it follows that we have in a vessel 
full of water a new principle of me- 
chanics and a new machine for multi- 
plyingforces to any degree we choose." 
Since the pressures on the two pistons are proportional to their 
areas, the mechanical advantage of a hydraulic machine of 
this sort is equal to the ratio of the area of the larger piston to that 
of the smaller. 

With regard to the work done, it should be noted that if in the 
example just mentioned the small piston is pushed through a dis- 
tance of 1 cm, the large piston will be displaced through only 0.01 cm, 
because the liquid forced out of the small cylinder into the larger has 
to spread over an area 100 times as great. Therefore, if we mul- 
tiply the forces by the corresponding displacements, we find 
that the resulting amounts of work are equal. It can easily be 
shown that this is true no matter what areas the pistons have, and 
no matter what the force and displacement of the smaller piston is; 
so that every hydraulic machine conforms to the general law of 
machines (c/. Art. 76). 

The principle of Pascal is extensively applied in a class of 
machines of which the hydraulic press. Fig. 70, is a type. It con- 
sists of a large, strong cylinder connected by a pipe with a force 
pump (Fig. 71). The piston of the large cylinder has an area many 
times larger than that of the pump. The pump piston is worked 



by means of a lever and forces water or oil into the large cylinder, 
causing the large piston to rise. Thus a bale of cotton, or whatever 
substance is to be com- 
pressed, is squeezed be- 
tween the pressure head of 
the large piston and the 
heavy frame above it. Hy- 
draulic jacks, used for lift- 
ing very heavy weights, 
work on the same princi- 
ple; and hydraulic ele- 
vators are operated by the 
ordinary pressure in the 
city water mains. In this 
case, the large piston is 
connected with a system 
of pulleys by which the 
displacement and speed of 
the motion are multiplied. 
The same principle is ap- 
plied in operating drills 
and other tools by means 
of compressed air. 

104. Pressure Due 
to the Weight of a 
Fluid. Since every fluid 
has weight, it follows 
that every surface sub- 
merged in a fluid in equi- 
librium is affected by a 
pressure which is due to 
that weight alone. 

Let us suppose that 

/ ^ . Fig. 71. The Hydraulic Press Diagram 

a very thin plate havmg 

an area of 1 cm^ is placed horizontally in water at a depth of 1 cm. 

What is the amount of the pressure on its upper surface? 

Fig. 70. The Hydraulic Press 


It may be Sjeen from the diagram, Fig. 73, that the water which 
rests on this surface is a vertical sided column 1 cm^ in sectional 
area; and its volume is 1 cm'. But since the density of water is 
1 ^, the mass of this cubic centimeter of water is 1 gm, and 
therefore its weight is 980 dynes. Since this mass rests on the 
given surface, it must be evident that it exerts a pressure directly 

Fig. 72. Mining Coal with a Compressed Air Drill 

on it, and that this pressure is nothing more or less than its 
weight. If the plate is so thin that its thickness may be neglected, 
its under surface will be affected by an equal upward pressure, 
because at any given depth the pressure due to the weight of the 
overlyjng fluid will be transmitted undiminished in all directions, 
as stated in Pascal's principle. 



Fig. 73. 

Liquid Pressure is Proportional 
TO Depth 

If the plate were at a 
depth of 2 cm, the pres- 
sure on it would be 2 gm- 
foree or 1960 dynes, be- 
cause the weight of the 
overlying water column 
would be that of a volume 
of 2 cm', and so on. 

Again, if the liquid, 
instea^d of being water, 
with a density of 1 ^, 
were mercury, which has 
a density of 13.6 f^,, the 
weight on each square 
centimeter would be 13.6 
times as great as that of an equal column of water. If the liquid 
were alcohol, whose density is 0.8 f^3, the pressure that it would 

exert on the submerged surface would 
be only 0.8 as great as that exerted by 
the weight of water at the same depth; 
and so on for other liquids. 

Since the pressure due to the weight 
of the fluid will be transmitted with un- 
diminished force to all equal areas, the 
total force on any given surface from 
this cause would be directly proportional 
to the area of that surface. 

Finally, if instead of being horizontal, 
the surface were vertical or oblique, the 
total force transmitted to it by the 
weight of the liquid would be exactly 
equal to that which w^ould be exerted 
directly on it if it were horizontal, jpro- 
vided its center of Thass were at the same 
depth; because for every small portion 
of the surface having a depth greater than 
that of the center, there will be an equal 

Fig. 74. Tall Standpipe: 
Great Pressure 


portion having a depth that is less than that of the center by just the 
same amount; and therefore, the mean pressure on every such pair of 
small portions of the surface will b(3 equal to the pressure at the 
center of mass. The total force on the surface will therefore be 
the same as it would be if the surface were horizontal and at the 
depth of its center of mass. 

Thus the following general statements may be deduced from 
Pascal's principle: 

The force due to the weight of a liquid in equilibrium, exerted on 
any surface submerged in it, is 

(1) Directly proportional to the depth of the center of mass of 
the surface, the depth being measured vertically from the center of 
mass to the level of the free surface of the liquid. 

(2) Independent of the direction in which the surface is turned, 
provided its center is kept at the same depth. 

(3) Directly proportional to the density of the liquid. 

(4) Directly proportional to the area of the surface. 

These principles enable us to calculate the amount of the force 
exerted on any surface by the weight of a liquid in which it is 
submerged; we have only to apply this simple rule: 

The force du£ to the weight of any liquid, exerted on a surface 
submerged in it, is numerically equal to the product obtained 
by multiplying together the weight per unit volume of the liquid, the 
depth of the submerged surface, and its area, 

106. Free Level Surface of a Liquid. We can now understand 
why the free surface of a liquid in equilibrium is level, and why, 
in a system of communicating vessels, like a teapot and its spout, 
the liquid stands at the same level in all the vessels. For if the surface 
were higher at one place than at any other place, the liquid pressure 
there would be greater, and the liquid therefore would flow from the 
place of higher level to the places of lower level, until no part of the 
liquid were higher than any other. 

106. Oases. Returning now to gases, we can appreciate that 
all we have said of liquids applies equally well to gases, with two 
important exceptions. First, the pressure in a gas is not propor- 


tional to the depth, because gases are easily compressed, so that 
the lower portions are denser than the upper ones. Second, gases 
have no such thing as a free level surface, for they tend to expand 
indefinitely, so as to fill completely the vessels in which they are 

107. The Air Pump. The phenomena of atmospheric pressure 
were very thoroughly investigated by Otto von Guericke (1602- 
1686), am eminent engineer, and burgomaster of Magdeburg. 

Guericke began experiments on the vacuum by filling a cask 
with water, and then pumping the water out of an opening at the 
bottom. Finding this method very imperfect, he finally suc- 
ceeded in inventing a fairly efficient air pump, which, a few years 
later, was much improved by Boyle and Hooke, in England. 

108. The Magdeburg Hemispheres. Guericke made many 
experiments, one of the cleverest of which was that of pumping the 
air out of a pair of hollow iron hemispheres having smooth rims 
which fitted very accurately together. When the air was pumped 
out of these, it was found, as Guericke expected, that great force 
must be exerted in order to separate them; because the pressure due 
to the weight of the air above them held them together. Since the 
force required to overcome this excess of external pressure is con- 
stant, no matter in what direction the axis of the hemispheres is 
turned, the experiment proves that at any given place the pressure of 
the atmosphere acts with equal force in all directions. 

This experiment was made by Guericke in the presence of 
Emperor Ferdinand II and the Reichstag, with hemispheres 1.2 
feet in diameter. The force of sixteen horses was required to sepa- 
rate them. Of course eight horses would have done as well if he 
had attached one of the hemispheres to a wall or post, but the 
dramatic effect might not have been so great. Plate V is a photo- 
graph of the picture that appeared in Guericke's book. 

109. Density of Air. Guericke was the first to demonstrate 
that air has weight by pumping some out of a hollow globe instead 
of forcing it in as Galileo did. He used a vertical tube of water 


as a barometer or vacuum gauge. The density of air has been 
very accurately determined by weighing in accordance with the 
method of Guericke, and is .001293 ^3, at 0° Centigrade and 
76 cm barometer pressure. 

Since we have learned that the pressure of the Jitmosphere is 
about one kilogram-force on each square centimeter, the question 
naturally arises as to how we are able to withstand so great a pressure 
on our bodies. The reason is that our blood and tissue cells 
contain air at the same pressure. The presence of this air can be 
demonstrated in an experiment with the "hand glass." This is a 
receiver which fits on the plate of the air pump, and has an opening 
at the top to which the palm of the hand can be fitted air tight. 
When the air is pumped out of the receiver, the hand is not only 
pushed down with great force by the weight of the overlying air, 
but also the fleshy part of the palm swells out and extends 
through the opening into the receiver. This is because the pressure 
of the outside atmosphere has been removed from that part of the 
hand, and the air within the hand, being now freed from this pressure, 
expands and distends the cells in which it is confined. It is for this 
reason that aeronauts and mountain climbers often suffer great in- 
convenience, for, as they ascend, the pressure of the atmosphere 
diminishes so rapidly that the blood is forced to the surface by the 
pressure of the air within their tissues. This unbalanced pressure 
puts an unwonted strain on the blood vessels, which often causes 
some of them to burst. 

110. Theory of Pumps. We are now in possession of all 
the information needed for explaining why a pump acts, as 
well as how it acts. When the piston P; Fig. 75, is with- 
drawn, it removes the atmospheric pressure from the fluid in the 
cylinder C; and the fluid is pushed into the suction pipe by the 
atmospheric pressure outside. Therefore the resultant force tend- 
ing to push the fluid into the cylinder is equal to the atmospheric 
pressure diminished by the weight of the fluids in the suction pipe. 
If the atmospheric pressure exceeds this weight, the inlet valve iv 
will be pushed open, and some of the fluid will be pushed from the 
suction pipe into the cylinder. Meanwhile the pressure due to the 



weight of the fluid in the outlet pipe, plus the atmospheric pres- 
sure, keeps the outlet valve ov closed, and prevents the reentrance 
of any fluid from the outlet pipe. When 
the piston is pushed in, it exerts on the 
fluid m the cylinder a pressure which closes 
the inlet valve, opens the outlet valve, and 
pushes the fluid out through it. 

Thus every double stroke removes a 
volume of fluid which is very nearly equal 
to the area of the piston multiplied by the 
length of the stroke. If the fluid to be 
pumped is air or any other gas instead of 
to be pumped into or out of a closed 

Fia. 75. Pump Diagram 

a liquid, and if it is 
vessel, the pressure in the 
closed vessel is mainly caused, not by the weight of the confined 
gas, but by its elasticity. 

111. Archimedes's Principle. Another important corollary 
deducible from Pascal's principle, is known by the name of Archi- 
medes. Let Fig. 76 
represent a vessel filled 
with liquid to the level 
ab, in which is sub- 
merged a rectangular 
solid cdef, and let us 
find the resultant force 
due to the weight of 
the liquid and tending 
to move the solid. 

The resultant of all 
the horizontal forces is 
zero, because every 
such force is opposed 
by an equal and oppo- 
site force at the same 
depth on the opposite side. But how about the vertical forces? 
The force on cd is equal to the weight of a column of liquid repre- 
sented by cdhg, acting downward; and the force 'on fe is equal 

Fig. 76. Buoyant Force Equals Weight 
OF Water Displaced 

128 / PHYSICS 

to the weight of the column of liquid jehg, transmitted as described 
by Pascal's principle, and acting upward. The resultant of these 
two forces, and hence of all the forces, is equal to their arithmetical 
difference, which evidently is equal to the weight of a volume of the 
liquid represented by Jecd, This volume is that of the body, and 
therefore of the liquid displaced by it. 

Archimedes of Syracuse (287? — 212 B.C.), who was the greatest 
natural philosopher that lived before the time of Galileo, must have 
known much of what we have learned about the equilibrium of 
liquids, for he discovered the principle that we have just reached 
and announced it substantially as follows : 

A body immersed in a fluid is huoyed up by a force that is equal 
to the weight of the fluid displaced by it. 

The foregoing argument for the principle of Archimedes does 
not depend for its conclusion on the kind of fluid, npr on the 
depth to which the body is submerged. It has also been shown to 
apply to bodies of all shapes and sizes. 

112. Floating Bodies. By Archimedes's principle, we can 
easily predict whether a body will float or sink in any fluid, whether 
that fluid be a Hquid or a gas. Thus, if the weight of the body is 
greater than that of an equal volume of the fluid, the body will sink 
to the bottom, of the fluid; if the weight of the body is less than that 
of an 'equal volume of the fluid, the body, if submerged, will float 
upward; if the weight of the body is exactly equal to that of an equal 
volume of the fluid, the body will remain wherever it is placed within 
the fluid. 

From the foregoing principles it follows that if a body which is 
lighter than the same volume of a given fluid be placed in that fluid, 
it will rise or sink (depending on where it is placed), and will 
come to equilibrium when it displaces a volume of the fluid that 
weighs exactly as much as it does. The buoyant force will then 
exactly balance its weight. This special case of the application 
of the principle of Archimedes is known as the principle of flo- 

The principle of Archimedes describes implicitly the behavior 
of a boat or a balloon. The more heavily a boat is loaded, the 



deeper it will sink. Why? Its gross displacement is the weight 

of the maximum volume of the water that it can safely displace. 

The weight of its maximum safe load is evidently the difference 

between its weight and its gross 

displacement. A very great load 

necessitates a correspondingly 

large displacement, which tends 

to diminish the speed that it can 


Boats are made practically un- 
sinkable by reserving a sufficient 
amount of the interior space for 
separate water-tight compartments, 
so that the boat can not sink by 
taking in water unless a number of 
the compartments are punctured 
at the same time. Submarine 
boats are made to sink by letting 
water into their compartments, and 
are made to rise by forcing it out 
with strong pumps. 

The load that a balloon, Fig. 
77, can support is equal to the 
weight of air displaced, dimin- 
ished by the sum of the weights of 
the balloon, car, rigging, con- 
tained gas, and ballast. If the aeronaut wishes to ascend, he 
diminishes the gross weight of the balloon by throwing out ballast. 
If he wishes to descend, he diminishes the volume of the balloon 
by opening a valve at the top and letting some gas escape. This 
diminishes the buoyant force. Why? 

Fig. 77. Santos Dumont's 

113. Determination of Density. Among the important appli- 
cations of the principle of Archimedes are several methods of de- 
termining density. The following example illub'trates one of these. 
A piece of rock weighs 25 gm. When suspended from the balance 
pan so as to be wholly submerged in pure water, it is balanced by 



15 gm. The buoyant force of the water on it is equal to its appa- 
rent loss in weight, or 10 gm. We have learned in Art. 32 that 
1 cm' of water has a mass of 1 gm; we know, therefore, that the 
volume of the 10 gm of water is 10 cm'. Since the volume of the 
rock is the same as that of the displaced water, or 10 cm', and 
since the mass of the rock was found to be 25 gm, its density is 

mass __ z5 gm _ « ^ gm 
volume 10 cm' * cm'* 

Archimedes used this method in determining for Hiero, king 
of Syracuse, whether some gold, which he had furnished to an 
artisan to be made into a crown, had been partly 
retained by that worthy, and the weight made up 
by alloying with baser metal. 

When we are possessed of the knowledge 
gained by the experimental researches of Galileo, 
Torricelli, Pascal, and their eminent contempo- 
: raries, it seems easy enough for us to under- 
stand and apply the principle of Archimedes; 
but when we remember that Archimedes lived 
nineteen centuries before these men, we can not 
> but marvel at the genius which enabled this 
great philosopher to think so clearly about this 
principle and its applications to liquids. 


FiQ. 78 
Boyle's Tube 

114. Boyle's Experiments. Among those 
scientists of the seventeenth century who con- 
tributed so largely to our knowledge of fluids 
was Robert Boyle of England (1627-1691). His 
greatest discovery, that of the law that goes by his 
name, was made during an investigation upon the elasticity of air. 
In order to find out what elastic force compressed air was able 
to exert, and what would be the effect of increased external pressures 
on the corresponding volumes, Boyle provided a tube. Fig. 78, 
"which, by a dexterous hand and the help of a lamp, was in such 
manner crooked at the bottom that the part turned up was almost 
parallel to the rest of the tube." The shorter leg was closed and 
the longer open. 


He started with the column of confined air 12 inches long, and 
with the mercury at the same level in both legs of the tube. Since 
the columns of mercury in the two legs then balanced each other^ 
the pressure on the confined air was simply that of the atmosphere. 
(Why?) Reading his barometer, he found this atmospheric pressure 
equal to 29 inches of mercury. More mercury was then poured 
in, till the column A was 29 inches long; the pressure on the 
confined air was therefore twice 29 inches. (Why?) When the 
confined air column was under this pressure, its length e was found 
to be 6 inches. Thus, when the pressure had been doubled, the 
volume was reduced to one-half. Another 29 inches of mercury 
reduced the volume to 4 inches, or one-third; and so on. 

115, Boyle's Law. As a result of extended experiments, 
therefore, Boyle announced the following law: The volume of a 
given mass of gas, at a constant temperature, varies inversely as 
the pressure that it supports. 

In symbols, if V and F' represent any two volumes of a certain • 
mass of gas at a constant temperature, and P and P' the corre- 
sponding pressures, Boyle's law is represented by the equation 

V P' 

f, = ^orrP = F'P'. 

The latter form of the equation shows that the products obtained by 
multiplying together each volume and its corresponding pressure 
are all equal. Therefore, calling this constant product K, we may 
represent Boyle's law by the equation VP = K and express it in 
ordinary language as follows: 

At a constant temperature, the product of the numbers represent- 
ing the volume and pressure of a given mass of any gas is a constant 
quantity , This law has been verified by a great many experiments, 
and is found to be approximately true for all gases within certain 
^ide limits; but at certain temperatures and pressures for each gas 
the law fails, notably when, on account of great compression, or 
low temperature, or both, the gas is about to liquefy. 

Aeriform bodies, like air or steam, are classified as perfect 
GASES when they act in accordance with Boyle's law, and as 
VAPORS when they do not so act. 


The graphical representation of Boyle's law is very interesting, 
and of great assistance in the solution of problems connected with 
the clastic pressures of gases in the cylinders of engines using the 
energy of steam, gas, or compressed air. 


1. The average pressure, at sea level, of the atmosphere bal- 
ances the weight of a column of mercury 76 cm high. This pres- 
sure is equal to 1032.84 grams-weight per square centimeter, and 
is called one atmosphere. 

2. The barometer is used to measure atmospheric pressure. 
It is applied: 1, in weather observations; 2, in determining eleva- 
tions; 3, in many experiments with gases. 

3. A pressure exerted on any portion of a fluid enclosed in a 
vessel is transmitted undiminished in all directions, and acts with 
equal force on all surfaces of equal area, in directions perpendicular 
to those surfaces. (Pascal's principle.) 

4. Pressure due to the weight of a liquid in equilibrium is pro- 
portional to its depth and to its density. 

5. The pressure due to the weight of a gas is not proportional 
to its depth. • 

6. Liquids have a free level surface; gases do not. 

7. Gases tend to expand indefinitely. 

8. A body immersed in a fluid is buoyed up by a force equal 
to the weight of the fluid displaced. (Archimedes's principle.) 

9. When a body is submerged in water, the number of gm of 
water displaced by it is equal to the number of cm' in its volume. 

10. The volume of a given mass of any gas at constant tempera- 
ture is inversely proportional to the pressure that it supports. 
(Boyle's law.) 


1. Describe the simple mercurial barometer, as arranged in the 
experiment of Torricelli. 

2. Why is the mercury column thus upheld shorter than the column 
of water in the suction pipe of a pump? 

3. Is there any air in the space above the mercury in a Torricellian 


4. Explain briefly why a falling barometer indicates stormy weather, 
and a rising barometer, fair weather. 

5. Explain how a barometer may be used for measuring altitudes 
above sea level. 

6. Why is it that if a balloon were ascending imiformly a barometer 
column would fall faster near the surface of the earth than it would at 
a greater altitude? 

7. Show how a mechanical advantage may be obtained from a body 
of fluid, as in the case of a hydraulic press. Show that a hydraulic 
machine conforms to the general law of machines. 

8. State four facts about the force due to the weight of, a liquid in 
equilibrium, and show how they are deducible from Pascal's principle. 

9. Suggest some experiments by means of which these fo\ir facts 
might be verified. 

10. State a rule, derived from these facts, for determining the 
total force exerted by the weight of a liquid on a surface submerged in 
it. In this rule, how is the depth to be measured? 

11. By means of a suitable diagram, explain why the free surface 
of a liquid in equilibrium is level. 

12. Describe the experiment of the Magdeburg hemispheres. Tell 
what it proves, and how it proves it. 

13. Describe Guericke's method of proving that air has weight, 
and contrast it with Galileo's. 

14. Explain why animals can withstand the great crushing force 
of the atmosphere. 

15. Draw a sectional diagram of a force pump, and fully explain 
both how and why the fluid is propelled through it. 

16. Discuss the application of the principle of Archimedes to a 
boat, and to a balloon. 

17. Explain how to calculate the load that a given boat or balloon 
can support. 


1. On a mountain the barometer reads 45 cm; what is the pressure 
of the atmosphere there (a) in ^^^ (6) in ^|5? 

2. When the atmospheric pressure supports a column of mercury 
75 cm high, how high a column of water will it support? How high a 
column of alcohol? Take the densities, as: mercury, 13.6; water, 1; 
alcohol, 0. 8. 

3. A plunger whose cross-sectional area is 4 cm^, is pushed into a 
cylinder full of oil with a force of 5 X 10* dynes; what pressure in -^I^ 
must be sustained by the walls of the cylinder? If the end of the cyl- 
inder has an area of 300 cm^, what is the total force exerted on it? 


4. The pump plunger A, of a hydraulic press, Fig. 71, has 
an area of 5 cm^ and the ram B an area of 1000 cm^; what is tlie 
mechanical advantage? How many kilograms-force must be applied 
to the plunger in order that the pressure head of B shall exert a total 
force of 9 X 10* Kg-force? 

5. When a man presses down on the lever of the press of problem 
4 at a distance of 60 cm from the fulcrum, what is the mechanical ad- 
vantage of the lever if the plunger is 10 cm from the fulcrum? What 
force must the man exert in order that the force on the plunger may . 
be 450 Kg-force? What is the total mechanical advantage of the press 
when the lever is used? 

6. When the standpipe in Fig. 74 contains water to the height 
of 30 m, what pressure in ?ni^°p5 does the water exert at the bottom? 
If one of the steel plates on the side there is 3 m long and 1.8 m high, 
what total force must it withstand? 

7. Find the pressures in in^_2r£? ©n the sides of the standpipe 
at depths of 1 m, 2 m, 5 m, 10 m, 20 m, 25 m, and, using any con- 
venient scale, plot a graph showing the relations of pressure to depth 
of the water. Does this graph suggest to you anything about the 
relative thicknesses that the steel plates' must have at these various 
depths in order not to burst? 

8. Fig. 79 represents an experiment devised by Pascal to verify 
the conclusions stated in Art. 104. The bottom is held on to each of 

the three vessels in turn by the pull of the 
cord only. If the distance from the pointer 
to the bottom is 10 cm, and the area of 
the bottom is 50 cm^, how many gm-force 
on the other balance pan are required to 
hold the bottom on when the cylindrical 
vessel is filled up to the pointer? How 
many for the wide-topped vessel? How 
many for the narrow-topped vessel? 
Would the result be the same if a vessel 
Fig. 79 having any other shape were used, pro- 

vided the other conditions remained the 

same? Explain how this apparatus may be used to verify statements 

1, 3, and 4 of Art. 104. 

9. Taking 1 -^^^^® as the pressure of the atmosphere and 36 

cm as the diameter of the Guericke hemispheres, Plate V, calculate 
the force with which they were held together, assuming that he got 
a perfect vacuum inside them. N. B. — The surface to be used is the 
area of greatest cross-section, not the surface of the sphere. (Why?) 
The latter would give the crushing force; calculate it. 



10. A balloon contains 300 m^ of illuminating gas, which weighs 
One m'of air weighs 1.3 Kg; what weight, including its own, 

will the balloon support? 

11. A canoe weighs 75 lb., 1 ft' of water weighs 62.4 lb. How many 
ft^ of water must the boat displace when it is carrying two persons, 
weighing together 240 lb.? 

12. The weight of a steamer is 6000 tons, and its gross displace- 
ment is 10,000 tons; what load can it carry? 

13. Fig. 80 represents a syphon. Suppose the atmospheric pressure 

is 103 


and that EA == 10 cm, DB = 20 cm. With how many 

At B? 

What is the result- 
What will 

^™cm2^^ ^^^^ ^^® water press down at A ? 

ant pressure in the direction ACB? In the direction BCA? 

the water do? Would the syphon work on a moun- 
tain top? In a vacuum? Over how great a height 

can water be raised by it when the barometer 

stands at 75 cm? 

14. Fig. 81, the cylinder C is hollow and has a 

capacity of 100 cm^. P exactly fits it. P and C 

are balanced, as shown, but without any water in 

the vessel. The water is then placed in the vessel 

under P; will it remain submerged? If water is 

poured into C until equilibrium is restored, how 

many gm will be required? How many cm'? 

15. Given: the mass of a 
piece of glass = 50 gm, the 
weight of the glass in water 
= 30 gm, the weight of the 
glass in gasoline = 36.26 gm. 
Required : the volume of the 
glass, its density, the mass of 
gasoline that has the same vol- 
ume as the glass, and the den- 
sity of the gasoline. 

16. A piece of wood having a mass of 37.5 gm is attached to a 
piece of lead whose mass is 166.5 gm. The weight of both, when sub- 
merged in water, is 139 gm. The lead alone weighs in water 151.5 
gm. Find: (a) the volume of water displaced by both together; (b) the 
volume of the lead; (c) the volume of the wood; (d) the density of the 
wood; (e) the density of the lead. 

17. If you hold your finger at the outlet valve of a bicycle pump 
when the pjston is at the top of the cylinder, and if the piston is then 
pushed half-way down, what is the pressure on your finger in 5l:^^£® 

if the pressure gf the atmosphere is 1 ^j? What are the pressures when 

Fig. 80 

Fig. 81 












50 3 










the piston is i, A» J o^ *he way down? If you pump up your tire until 
the pressure in it is doubled, by how much is the density of the air 
in it changed? 

18. Appended are some of the data 
obtained by Boyle in his experiment 
(Art. 114) : Choosing a convenient scale, 
plot the lengths of the air columns as ab- 
scissas, and the corresponding pressures 
as ordinates. The graph will then rep- 
resent the relation PV = const. What 
does the graph show about the pressure 
when the volume becomes very large? 
What about the volume when the pres- 
sure becomes very large? 


1. By means of a rubber tube connect a bubble pipe with the gas 
fixture, let the gas blow soap bubbles for you, and see what they will 
do. Can you explain their behavior? 

2. Visit the water works and find out all you can about the pump- 
ing engines and the pumps. How is the pressure regulated by means 
of air chambers connected with the inlet and outlet pipes? Find out 
whether there is a standpipe connected with the works, and if so, 
what its use is. 

3. Can you explain how your bicycle pump works? Find a com- 
pressed air tool at work (a riveter on a steel framed building or bridge, 
or a cutting tool at a marble works), and learn what you can about 
how they work. 

4. When a sail-boat is tipped from the position in which the deck 
is horizontal, what moment of force does most of the tipping? What 
moment tends to restore it? Investigate this interesting and important 
problem of the stability of a boat and find out what you can about it. 

5. Consult a book on physiology in which the action of the heart 
is described. Can you understand wherein it resembles a force pump? 
In what essential detail does its action dififer from that of an ordinary 

6. If you are interested in the air ship shown in Fig. 77, read Santos 
Dumont's book on My Air Ships (N. Y. Century Co., 1904)." 


116. Heat and Work. In the preceding chapters we have 
studied the operation of the locomotive and of other machines and 
learned of motions and mechanical efficiencies. How is it with 
steam engines and gas engines? Every one is familiar with the 
fact that every engine consumes fuel in some form, and that 
without the fuel it will not move at all. Hence, all engines 
must in some way derive the energy with which they do work 
from the fuel that they bum, i.e., an engine is simply a device 
for converting heat energy into mechanical work. 

The questions that arise in connection with the conversion of 
heat into mechanical work are many and interesting. Thus, 
how can we measure heat? How do bodies change when they 
are heated or cooled? What of the process of converting water 
into steam? Is heat absorbed in this operation? Does steam act 
as a gas and obey Boyle's law, or does it act differently? How 
is the heat transferred from the fire to the water in the boiler 
and from the water into steam? Is there any definite relation 
between heat and mechanical work? Is the efficiency of a steam 
engine high or low, and how is it determined? On what factors 
does the efficiency of such an engine chiefly depend? 

117. Heat Sensations Xrnreliable. Perhaps the most familiar fact 
about heat is that some things feel hot to our touch while others 

,feel cold. Yet our ability to judge how hot a body is depends on 
a number of varying circumstances, and at best is limited. For 
example, if we take three basins of water, one hot, one lukewarm 
and one cold, and place the right hand in the hot water and the 
left in the cold, and then transfer both hands to the lukewarm water, 
this latter will seem cold to the right hand and hot to the left. 
Hence, we can not rely on our sense of touch for accurate informa- 




tion concerning differences of temperature. What, then, may we 

118. Galileo's Thermometer. The first to give any scientific 
answer to this question was Galileo. He blew a bulb on the end 
of a glass tube of small bore and after slightly warming the bulb 
placed the end of the tube in a vessel of colored water .(Fig. 82). 
When he warmed the bulb with his hand the liquid 
in the tube moved downward, showing that the air 
in the bulb expanded. Also conversely, when he 
cooled the bulb the liquid in the tube moved upward, 
showing that the air in the bulb contracted. He thus 
showed that air expands when it is warmed and con- 
tracts when it is cooled; and he suggested the use of 
this property of air for detecting small differences of 

It is probably not necessary to state that Galileo's 
suggestion has been universally adopted for scientific 
work, although the instrument which he devised is 
practically useless as a thermometer, because the liquid 
in it is exposed to the pressure of the atmosphere. Since, 
as we learned in the last chapter, this pressure is 
always changing, the small column of liquid moves 
Fig. 82 when the atmospheric pressure changes as well as when 
Thermom- the temperature changes. Therefore, we can not be 
sure that a given position of the liquid indicates the 
same temperature at different times. Fortunately, allowance can 
be made for the error thus introduced, provided the barometer 
is observed at the same time with the thermometer, so that the 
pressure on the air in the bulb is known. But even so, how may 
we determine the amount of the change in temperature correspond- 
ing to a given motion of the liquid in the tube? 

119. The Temperature Scale. We can not answer this question 
until we have adopted a scale of temperature, and defined the units 
in terms of which we shall measure differences of temperature. In 
order to establish such a scale it is necessary to have some fixed 

HEAT 139 

temperature which may be used aS the zero from which to count, 
and also to have some unit difference of temperature. It has been 
found convenient to adopt the temperature at which ice melts as 
the ZERO temperature; hence, in scientific work this is called 
a temperature of zero degrees. 

In order to determine a unit ' difference of temperature, we 
•must select some other fixed temperature, and then define the 
interval between the zero and this other temperature as a certain 
number of degrees. The second fixed temperature that has been 
adopted by scientists is that of water boiling at normal barometer 
pressure, i.e., 76 cm. The interval between the temperature of 
melting ice and that of boiling water has been divided into a hun- 
dred equal^ temperature intervals called degrees. Another temper- 
ature scale, called Fahrenheit's, is in common use, but the one just 
defined is generally used in scientific work. Since, in defining 
this scale, the fundamental temperature interval is di>dded into 
one hundred equal parts called degrees, it is called the centi- 
grade SCALE. Temperature degrees in this scale are denoted by 
the symbol °C. For example, 40° C. means forty degrees of the 
Centigrade scale. As the temperature does not involve either gm, 
cm, or sec, it has no symbol in the terms of these units. 

Having defined our temperature units, we are now in a position 
to put a scale on our thermometer. This is done by placing the 
instrument in melting ice, marking the position of the drop of 
liquid in the tub», then placing the instrument in the steam over 
boiling water, and marking the position of the drop of liquid when 
it has become stationary. The interval on the tube between 
these two marks is now to be divided into one hundred parts 
representing equal temperature intervals. 

120. Change of Volume at Constant Pressure. Let us now 
ask what the relation is between the volume of the air in the bulb 
and an" increase in temperature of 1°. This relation has been deter- 
mined experimentally with great accuracy, and it has been found 
that when a given mass of gas is heated from 0° to 1° its volume 
increases ^, when heated from 0° to 2° its volume increases gfg, 
of its volume at 0°, etc., i.e., for every change of 1° in tem^perature, 


the corresponding change in volume is 273 of the volume at 0°. 
This ratio has been found to be the same for all gases and for all 
changes of 1° in temperature. It is called the coefficient of 
EXPANSION OF GASES. Since the measurements by which these 
facts were first established were made by Charles and Gay Lussac, 
this relation is known as the law or Charles and Gay 

If we let V represent the volume of the gas at any temperature 
/, and Vq its volume at 0° C, then, since the final volume (V) is 
equal to the volume at 0° (Vq), plus the increase in volume (273 ^o)> 
this law is expressed analytically as follows: 

^=^«0 + 2f3> 

1 . ' V 

On factoring out ^;^, this equation becomes V = — -^ (273 + i). 

Since r~ is a constant for any mass of gas, we see that at constant 

pressure the volume of a given mass of gas is proportional to 
(273 + t). 

It is to be noted that this equation accounts for changes of 
volume due to changes of temperature only; and hence, in stating 
this equation it is assumed that the pressure of the gas remains 

121. Change of Pressure at Constant Volume. We have 
just found how the volume changes when a gas is heated at constant 
pressure; let us now try to find out how the pressure varies when 
the gas is heated while its volume is kept constant. 

In order to do this we must add to the thermometer of Galileo 
a device for governing and measuring the pressure in the bulb 
This device usually consists of a rubber tube K, Fig. 83, which is 
fastened at one end R to the glass tube of the thermometer and 
at the other R to another similar piece of glass tubing. This 
rubber tube is then filled with mercury until the mercury appears 
above both its ends. The instrument as thus arranged is called 
an AIR THERMOMETER. It will be readily seen that by raising or 



lowering the free end ii' of the rubber tube we can increase or 
diminish the pressure of the air in the glass bulb. 

If the glass bulb is now heated, the air within it will expand 
and depress the mercury at the end R of the rubber tube. To 
compress the air to its original volume, the other end of that tube 
must be elevated until the mer- 
cury in the thermometer re- 
turns to its former level d. 
Thus, the increase in pressure 
produced by the rise in temper- 
ature is balanced by the pressure 
due to the weight of the mercury 
in column h; and the total pres- 
sure may be found by adding 
that of the column h to that of 
the atmosphere as read from 
the barometer (c/. Art. 115). 
What is the relation between 
the increase in temperature 
and that in pressure when the 
volume of gas is kept constant? 

We can find the answer to 
this question by measuring the 
changes in temperature and the 
corresponding changes in pres- 
sure. This has been carefully 
done, and the relation is found 
to be similar to that between 
change of volume and change 
of temperature at constant pres- 
sure. It is stated as follows : when a given mass of gas is heated 

at constant volume, the pressure increases - — of the pressure at 0° 

for every change of 1° in temperature. This is true also for all 
gases and for all their changes of temperature. 

If we let P represent the pressure of a given mass of gas at any 
temperature t, and Pq its pressure at 0° C, then, since the final 

Fig. 83. Air Thermometer 


pressure (P) is equal to the pressure at 0° (Pq), plus the increase 
in pressure ( ^— - P^Y the result is expressed analytically as follows: 


Or, factoring out the 2^3' ^ = ^^ (^73 + 0- 

We can therefore find the value of t with the air thermometer 
by subi^titutirig the observed values of P and P^ in this equation, 

and solving it for /. Thus t =— ^ 273. 

Since -^—^ is a constant for a given mass of gas, we see that 

at constant volume the pressure of a given mass of gas is propor- 
tional to (273 -\- t). It will aid the memory to note that the equa- 
tions of this article, and those of Art. 120 are similar in form. 

The air thermometer is not a simple instrument to handle, 
therefore temperatures are generally measured by the ordinary 
mercury thermometer, but it must not be forgotten that the air 
thermometer is the standard to which the mercury tJiermometers are 
all referred. 

122. When Pressure, Volume, and Temperature Change. 
Now, it has been shown in Art. 120 that, at constant pressure, 
273 + t is proportional to the volume V. It now appears that at 
constant volume this quantity is also proportional to the pressure P. 
Therefore it follows that 273 + t is proportional to the product of 
,VandP; i.e., PV = Constant X (273 + t). The numerical value 
of this constant depends on the density of the gas and the units 
in which the quantities are expressed. For solving most gas prob- 
lems we may express this relation in a more convenient form. 
Thus if V is the volume of a mass of any gas at <° C. and pressure 
P, and F' its volume at temperature f° and pressure P', then 

VP _ 273 + < 
V'P' 273 + f ^^ 

This equation expresses the relations of pressure, volume, and 
temperature for gases. It means that the product of the volume and 

HEAT 143 

pressure of a given mass of gas is directly proportional to 273 + its 

123. Absolute Temperature. Since the quantity PV is not 
proportional to the temperature as measured on the Centigrade 
scale, but is proportional to 273 plus that temperature, it is convenient 
for work of this kind to conceive that the zero temperature is placed 
at —273°. This new zero of temperature is called the absolute 
ZERO, and temperatures measured from it are called absolute 

Therefore 273 + t represents the absolute temperature. 

124. Expansion of Solids and Liquids. Thus far we have 
studied the changes in gases caused by changes in their tempera- 
tures. Do solids and liquids expand when they are heated, and 
contract when they are cooled? Everybody knows that they do. 
Just as in the case of gases, the volume of every solid and liquid 
is changed by a certain fraction of itself for every degree that the 
temperature changes. This fraction is called the coefficient of 
cubical expansion. Unlike gases, however, each liquid and 
solid has its own coefficient of expansion which is characteristic 
of it. 

If V represent the volume of any solid or liquid at a tempera- 
ture t, Vq its volume at 0° C, and c its coefficient of expansion, 
then F = Fo (1 + ct) (cf. Art. 120). 

Similarly for solids when the change in length only is important, 
the fractional change in length for 1° C. is called the coefficient 
OF linear expansion. If a represent this fraction, L the length 
of the solid at any temperature /, and Lq the length at 0° C, then 
i = Zo (1 + at). 

These coefficients are needed for the solution of many impor- 
tant problems which arise in everyday life because of the expansion 
and contraction due to changes in temperature. For example, 
when railroads are constructed in winter, spaces must be left 
between the ends of the rails to allow for the expansion in summer. 
Long span bridges must have their ends placed on rollers; and 
provision must be made in a hot water heating system for the 
expansion of the water and of the pipes. 


125. Measurement of Heat. We have defined units in 
which we may measure temperature, but does the determination 
of the temperatures give us any information about the amount 
of heat? May not a small body and a large body have the same 
temperature but contain very different amounts of heat? Hence, 
in order to answer the question as to the quantity of heat absorbed 
by an engine or by any other device or body, we must make a 
further definition as to the units in which this heat is to be meas- 
ured. The unit universally adopted by physicists for quantities of 
heat is the quantity of heat absorbed by one gram of water when 
heated from 15° to 16° C This unit is called the gram calorie. 
Thus, if one liter (1000 cm) of water is heated until its temper- 
ature has risen through 100°, the quantity of heat thus imparted 
to it is 1000 X 100 = 100,000 gram calories. 

126. Specific Heat. But does it require equal amounts of heat to 
increase the temperature of all substances by 1°? Evidently not, 
for it is well known that it requires more heat to raise the tempera- 
ture of a gram of water through one degree than to raise the same 
mass of any other substance through the same temperature interval. 
In order to compare the heat capacities of different substances, 
it is, therefore, convenient to express their ability to absorb heat 
in terras of the heat-absorbing power of water. We may call this 
heat-absorbing power Specific Heat, and define it as the ratio 

heat absorbed by 1 gm of the given substance in warming 1° 
heat absorbed by 1 gm of water in warming 1° 
i.e., the specific heat of any substance is the number of calories 
required to warm 1 gram of it through 1° C 

The numerical value of the specific heat of any substance may 
be determined by experiment in a number of different ways. One 
of the simplest of these is illustrated by the following example: 
100 gm of aluminum clippings at 98° C. are stirred into 200 gm 
of water at 2° C; and the mixture comes to the temperature of 
11.5°. If h represent the specific heat of aluminum, the heat 
given up by it in cooling to 11.5° is 

hX 100 gm X (98°- 11.5°). 
Sp. Ht. X mass X change of temperature. 

HEAT 145 

The heat absorbed by the water in warming to 11.5° is 
1 X 200 gm X (11.5° - 2°). 
Sp. Ht. X mass X change of temperature. 

The heat absorbed by the water must be equal to that given 

out by the aluminum; so that we may form the equation, 

A X 100 X (98 - 11.5) = 1 X 200 X (11.5 - 2), or A = 0.225. 

In performing the experiment care must be taken to let as 
little heat as possible enter or escape, since it is assumed by the 
equation that none does so; and proper allowance must, in general, 
be made for the heat absorbed or emitted by the vessel that holds 
the w^ter. Vessels used for this purpose are called calorimeters, 
and are usually made of metal, and surrounded by a box designed so 
as to prevent heat from entering or escaping. * 

127. Steam. One other important phenomenon connected 
with an engine must be understood before we can determine its 
efficiency. This is the phenomenon of making steam. Heat 
is absorbed in this process; for a boiling kettle apparently 
ceases to emit steam shortly after being removed from the fire; and 
the greater the surface exposed to the air, the faster the water 
cools. But does water at any temperature ever cease emitting 
steam? If not, why do we have to heat it so hot in order to make 
it produce steam for use in the engine? Does water boil always 
at the same temperature, and what is the nature of the phenom- 
enon we call boiling? In order to ilnd answers to these questions, 
consider first a glass fruit jar of water exposed to the air. If 
this jar is placed in a warm room, v/hat will happen to the water? 
Suppose that the cover is sealed on to the jar, what will then happen 
to the water? Will it evaporate at all? If so, how much? If 
not, why not? Suppose that we place the open jar under the 
receiver of an air pump and exhaust the air, will the water then 
evaporate; and, if so, to what extent? Will it evaporate in the 
same room faster if it is hot than if it is cold? 

128. Evaporation. It is well known that water evaporates 
when left in open dishes, and that in the same room the evaporation 


is faster the hotter the water; hence, we will all agree that water 
passes into aqueous vapor or steam at all ordinary temperatures. 
We may also grant that the reason why water evaporates from an 
open dish but does not disappear from a closed jar, is that the room 
is so much larger than the jar, and hence is capable of holding 
very much more water vapor than the space in the jar can hold. 
But how much water vapor will a given volume hold, and is this 
amount the same at all temperatures? 

Elaborate experiments were necessary to determine these points, 
and the results show that water will always evaporate until its 
vapor exerts a certain pressure on the walls of the vessel con- 
taining it, and that this pressure can not, at a given tempera- 
ture, exceed a certain definite value. When the pressure reaches 
this maximum value the space in the vessel is said to be 
SATURATED, and hence this maximum pressure is called the pres- 
sure of the saturated vapor at the given temperature. This pres- 
sure is independent of the other contents of the vessel, and 
depends only on the temperature of the water and its vapor. The 
case is similar for other liquids. Hence, we see that evaporation 
takes plaice wherever there is an exposed surface of liquid, and 
continues until the vapor attains this maximum pressure of satu- 

Further, as long as any liquid remains in the closed space we 
can not increase the pressure of this saturated vapor, provided 
the temperature remains constant. Nor can we alter that pressure 
by changing the size of the space: for if we increase the space, more 
vapor is formed; if we decrease it, some vapor is condensed into 
liquid; and the pressure exerted by the vapor remains constant. 
Hence, we may say in general that the pressure exerted at a given tem- 
perature by saturated vapor in contact with its liquid is always the 
same. Of course, the pressure exerted by the saturated vapors 
of different liquids at a given temperature are not necessarily 
the same. 

A simple experiment will make these matters clear. In the 
barometer tubes 6, 6', 6", Fig. 84, the three mercury columns at 
first stand at the same height, depending on the amount of the 
atmospheric pressure at the time. Now, with a medicine dropper, 



^11 1 

ti ii 'j 

1 '■ ■'' 


insert a few drops of water under the end of the tube b\ tap 
the tube gently, if necessary, till the water rises to the top of 
the mercury column, and observe the result. The 
water evaiporates till the water vapor exerts its 
pressure of saturation corresponding to the temper- 
ature of the experiment. Manifestly, this pressure 
is measured by the depression ct of the niercury 
column. Similarly, a few drops of ether, inserted 
in 6", partially evaporate, and exert the pressure 
of saturated ether vapor corresponding to this 
temperature. This pressure is measured by the 
depression cs of the mercury column in b", and is 
seen to be greater than that of water vapor. If 
now a few drops of ether are introduced into the 
tube 6', the depression of the mercury there is seen 
to be equal to ct -{■ cs: i.e., the pressure of the ether 
vapor is the same as before, and so also is that 
of the water vapor. The total pressure, due to 
both, is simply the sum of the pressures which 
each would exert separately. So we see that the 
pressure of each is the same as it would be if the 
other were not present in the space. 

Now incline the tube b". The mercury compresses the ether 
vapor in it. The vapor now occupies a smaller volume, so some 
of it must have been condensed into liquid alcohol, but the mer- • 
cury still remains at the level s as before. The pressure cs of the 
ether vapor is therefore unchanged, although its volume has 
been much diminished. If the tube be held erect and lifted a 
little way (but not out of the mercury in the vessel), the mer- 
cury is seen to remain at the same level s. This time the volume 
has been increased, but the pressure remains constant, showing 
that some more ether must have evaporated to fill the space 
and exert its pressure of saturation. 

By warming or cooling the .vapor-filled space in either 6' or 6", 
it may easily be observed that the pressure of saturation is greater 
when the temperature is higher, and less when the temperature is 
lower. The relations between temperature and pressure of 

Fio. 84 



saturated vapor, as determined for water and for alcohol, are shown 
in the curves (Fig. 85). The abscissas represent temperatures, 
and the ordinates pressures in cm of mercury. The numbers on 
the vertical scale, when multiplied by 10, represent cm of pressure; 
those on the horizontal scale, when multiplied by 10, represent 
degrees C. What pressure does the line Ap represent? 


129. Boiling Point. From the curve W, tell what pressure is 
exerted by saturated water vapor at a temperature of 20°, of 50°, 
of 80°, of 100°. Do you note any relation between the pressure 

corresponding to 100° and the 
normal barometer pressure, 76 
cm? What pressure does alco- 
hol vapor exert at 20° (curve A), 
at 50°, at 78°? Do you note 
any relation between the pres- 
sure corresponding to 78° and 
the normal barometer pressure? 
Since at 76 cm pressure, water 
boils at 100° C. and alcohol at 
78° C, may we then define 
boiling point as the temperature 
at which the pressure of satur- 
ated vapor is equal to the sur- 
rounding pressure? Suppose the 
atmospheric pressure to be 42 
cm, as on the* top of Mt. Blanc, at what temperature would water 
boil there? At what temperature would alcohol boil there? 

Those who have* answered correctly the questions just asked 
will understand that the boiling point of any liquid may be defined 
as the temperature at which the pressure of its saturated vapor is equal 
to the surrounding pressure. Thus we see that a liquid may boil 
at almost any temperature, since a reduction of the external 
pressure lowers the boiling point and an increase in that pressure 
raises it. For water, this change in boiling point corresponding 
to a change of 1 cm in the barometric pressure is 0.37° C. For 
example, when the barometric pressure falls from 76 cm to 74 cm 

Fig. 85. Relation op PHessure of 
Saturated Vapor to Temperature 
for Water and Alcohol. 

HEAT 149 

a correct thermometer, placed in the steam of boiling water, will 
read 100° - 2 X .37 - 99°. 26 C. 

That the boiling point is the temperature at which the pressure 
of the saturated vapor is equal to the surrounding pressure may 
be readily appreciated from the common sense point of view. For 
it is plain that ebullition (i.e., boiling) is different from evaporation 
in that the steam escapes in bubbles from the midst of the liquid 
instead of from the surface only. Now if the surrounding pressure 
were greater than the pressure of the steam in these bubbles, the 
bubbles would be unable to expand and float to the surface. On 
the other hand, if the external pressure were less chan that of the 
steam composing the bubbles, the water would flash into steam 
instantaneously, as it sometimes does with explosive violence 
when a defective boiler gives way. 

130. Superheated Vapor. Let us now suppose that we have 
a very small quantity of water in a closed fruit jar at a given 
temperature. As has just been stated, the vapor will soon become 
saturated and will exert the pressure that corresponds to this 
temperature. If, now, we increase the temperature, more of the 
water will evaporate, and the pressure of the saturated vapor will 
increase to that corresponding to this higher temperature. Let 
us now suppose that at this higher temperature all of the water has 
been evaporated; what will be the effect of a further increase in 
temperature? The pressure will increase, of course, but will it 
increase as fast as it would if more liquid were present so that more 
vapor would be formed? Clearly not. Therefore, when a satu- 
rated vapor not in contact with its liquid is heated in a closed vessel, 
its pressure at the higher temperature is less than that which it 
would exert if it remained in contact with its liquid so that it con- 
tinued to be saturated. A vapor not in contact with its liquid and 
at a temperature higher than that corresponding to saturation, is 
said to be superheated. 

131. A Gas is a Superheated Vapor. Let us now consider 
how the pressure of a superheated vapor varies with the tempera- 
ture. So long as the vapor is superheated, none of it will condense, 


and there will be no liquid in the space, so that no more vapor can 
be formed therein and none of the liquid can exist. Therefore 
the changes in pressure produced by changes of temperature will 
not be complicated by evaporation and condensation. Hence we 
may surmise that such a vapor will behave very much like a gas. 
Experiment has shown that superheated vapors, when they are 
not too near the saturation point, do act in accordance with the 
gas laws of Gay-Lussac and of Boyle. We therefore conclude 
that a superheated vapor is really a gas. The converse of this con- 
clusion, i.e., that a gas is a superheated vapor, has been verified by 
the reduction to liquids of the so-called permanent gases, hydro- 
gen, oxygen, and nitrogen. 

132. Critical Temperature. We have seen that water can 
exist as a liquid at all ordinary temperatures. Therefore at all 
such temperatures superheated water vapor can be condensed 
to a saturated vapor or to a liquid by the application of pressure 
alone. On the other hand, the so-called permanent gases do not 
exist at ordinary temperatures as liquids, and we find that we can 
not, by any amount of pressure, condense them to liquids without 
also cooling them. The temperature to which a gas must be 
cooled before it can be converted into a liquid, is different for dif- 
ferent gases, and is called the critical temperature. The crit- 
ical temperature of water is 365^ C; that of air, — 140° C. Other, 
critical temperatures are: alcohol, 243° C; ether, 194° C; ammo- 
nia, 130° C; carbon dioxide, 31° C; oxygen, — 119° C; hydrogen, 

From what has just been said we learn that we can not condense 
a gas or a superheated vapor into a liquid by applying pressure 
only, if the temperature is above the critical value for that gas. This 
important fact is of far-reaching moment in the economy of nature, 
as a study of the preceding figures will show. We note that for 
water this critical temperature is high, so that at all ordinary tem- 
peratures water exists as a liquid : the same is true of alcohol. On 
the other hand, the critical temperature of air is very low. Hence, 
at all ordinary temperatures air is a superheated vapor or gas, 
and can not be liquefied by pressure alone: the same is true of hy- 



drogen and oxygen. Hence, we see why such low temperatures are 
required for liquefaction of these gases. We can readily under- 
stand how fortunate it is for beings organized as we are that these 
substances are so en- 
dowed; for if the crit- 
ical temperature of 
water were low, while 
that of air were high, 
we would know water 
at ordinary tempera- 
tures only as a gas, and 
air under like condi- 
tions largely as a liquid. 
The entire economy of 
nature would thus be 
overturned; for what could we do with liquid air to breathe and 
gaseous water to drink? 

Fig. 86. Water Changes to Invisible Vapor 
Which Condenses into Clouds 

133. Humidity. The relations we have just been studying 
act favorably to the maintenance of life on the earth in other impor- 
tant ways. Thus, the fact that at ordinary temperatures water and 
its vapor exist together shows us how it is possible for the water of 
the ocean to evaporate and be carried in the form of vapor over the 
land, to be deposited there as rain. We see also why there is always 
considerable water vapor in the air. This humidity of the air 

is an important factor 
in climate. Every one 
knows how oppressive a 
hot day is if the hu- 
midity is high; i.e., if 
the water vapor in the 
air exerts a pressure 
nearly equal to that of 
saturation at the tem- 
perature of the air. Under these circumstances it is plain that 
very little water can evaporate; and therefore we are not cooled 
by evaporation from our bodies. 

Fig. 87. Rain Clouds Deposit the 
Water on the Land 


134. The Formation of Dew. The formation of dew is a 
familiar phenomenon. Drops of water appear on the outside of a 
pitcher of ice water on a warm day, because the temperature of the 
pitcher is below that at which the water vapor in the air would 
be saturated; i.e., below the dew point. Thus, by dew point is 
meant the terrvperature to which the air mibst he cooled in order to 
bring the water vapor in it to saturation, so that condensation begins. 
Since the amount of water vapor in the air is of such great im- 
portance to climate, its detennination is an important part of the 
work of the Weather Bureau. What is termed relative humidity 
is the ratio of the actual pressure of the water vapor in the air to 
the pressure of saturated vapor at the same temperature. 

. 135. Latent Heat. Another important fact about the con- 
version of water into steam or into ice remains to be considered. 
If we place a thermometer in a vessel of water and heat it gradually, 
the thermometer indicates a gradually increasing temperature; 
but when the water reaches the boiling point, although we continue 
to heat it, the temperature remains constant until the change of state 
is completed. Thus, when water is boiling under any given pressure 
we can not raise the temperature of the water beyond the boiling 
point that corresponds to that pressure. But what becomes of the 
heat that is added after the boiling point is reached? It is used 
in converting the water into steam; so that energy is required 
to do this work. Is the quantity of heat thus required large? 
Experiment shows that it is large, for it is found that 536 gm cal 
are required to convert 1 gm of water at 100° into 1 gm of steam at 
the same temperature. Since this amount of heat seems to disap- 
pear in the process, it is called the latent heat of steam. Also, 
conversely, when steam condenses into water, it gives up its latent 
heat, every gram of steam returning its entire 53G gm cal. 

A similar phenomenon accompanies melting and freezing, 
but the amount of heat required' is not so great; thus it requires 
80 gm cal of heat to convert one gm of ice at 0° C. to one gm of 
water at the same temperature. Conversely, when water is frozen 
it gives up this same quantity of heat per gm. Every substance 
absorbs a definite amount of heat per gm while melting or evapo- 

HEAT 153 

rating and gives up this energy while solidifying or condensing, the 
amount thus transformed being different for different substances. 

136. Latent Heat is a Form of Energy. From the fore- 
going discussion it must be quite clear that latent heat is a form 
of energy, for heat energy is expended in doing the work of convert- 
ing the liquid into the vapoi: form, and is given up again as heat 
when the vapor is condensed into the liquid form. 

We can now * understand why it is that when a substance is 
vaporizing or condensing, or when it is liquefying or solidifying 
under a constant pressure, its temperature remains constant 
until the transformation is completed. For when heat energy. 
is doing the work of changing the state or internal condition of the 
substance, it can not at the same time be employed in raising the 
temperature. Conversely, when a mass of vapor is liquefying, 
each gm of vapor that condenses gives up its latent heat, so that 
the temperature of the liquid can not fall so long as any vapor 
remains to be condensed and supply it with heat. The case is 
the same when liquids solidify. 

137. Water and Climate. We can understand also why 
water is so important in regulating atmospheric temperatures, 
because its specific heat, and its latent heat of vaporization and of 
solidification are so great. When water is warmed, or changed 
from ice to water, or from water to vapor, it absorbs large quan- 
tities of heat, and so prevents the atmosphere's heating as rapidly 
as otherwise it would. Conversely, when it is cooled, or changed 
from water to ice or from vapor to water, it gives out large quan- 
tities of heat, so that the temperature of the atmosphere does not 
fall as rapidly as otherwise it would. Since water is evaporated 
in great quantities from the oceans and since some of it is then 
carried with the winds over the land to be there condensed, it serves 
the earth very much as a steam heating system serves our offices 
and dwellings. 


1. Zero temperature is that of melting ice. 

2. Unit temperature interval is the Centigrade degree. This 
is the j^ part of the interval between the temperatures of melting 
ice and water boiling at 76 cm barometer pressure. 


3. When the pressure remains constant, a given mass of gas ex- 
pands 2I3 of its volume at 0° C, for every increase in tempera- 
ture of 1° C. (Gay-Lussac's Law.) 

4. Absolute temperature is equal to 273° + Centigrade tem- 

5. When its volume remains constant, the pressure exerted 
by a gas is proportional to its absolute temperature. 

6. Unit quantity of heat is the gram calorie, i.e., the quantity 
of heat involved in changing the temperature of 1 gm of water 
1° C. Its symbol is gm cal. 

7. Specific heat of a substance equals the number of calories 
absorbed by. one gram of it in warming 1° C. 

8. The total number of calories absorbed or given off by any 
body during any change of temperature = specific heat X mass X 
change of temperature. 

9. Every saturated vapor exerts a pressure that depends on its 
temperature only. 

10. A gas is a superheated vapor. 

11. Gases can not be condensed into liquids by pressure, however 
great, at temperatures above the critical temperature. 

12. Water vapor is an important constituent of the earth's 

13. The dew point is the temperature at which the water vapor 
in the atmosphere becomes saturated. 

14. The relative humidity of the atmosphere is the ratio of the 
pressure of saturated vapor at the dew point, to its pressure at the 
temperature of the atmosphere. 

15. Heat is absorbed during the processes of melting and 
evaporation, and given out during the converse processes of solidifi- 
cation and condensation. (Latent heat.) 

16. 80 gm cal of heat become latent heat when one gm of ice 
melts; and 536 gm cal, when one gm of water evaporates at 
100° C. 

17. When a body changes state, the number of calories absorbed 
or given off by it is equal to the corresponding latent heat, multi- 
plied by the number of grams mass. 

18. Latent heat is a form of energy. 

HEAT 155 


1. How far can we rely on our sense of touch for information con- 
cerning temperature? Give some examples. 

2. What elements are necessary in determining a temperature 
scale, and how is the Centigrade temperature scale defined? 

3. Why is Galileo's air thermometer inaccurate? What device is 
employed to keep either the volume or the pressure of the air con- 

4. What other instrument must be used in connection with an air 
thermometer in determining temperature? Why? 

5. How much does a gas expand when heated 1°C.? What is Gay- 
Lussac's law? 

6. What do we mean by absolute temperature? 

7. Is there any relation between the pressure of a gas at constant 
volume and its absolute temperature? What is the relation? 

8. How do we define the unit quantity of heat? 

9. How do we compare heat quantities? What is specific 

10. Does water vaporize at all temperatures? 

11. If we have water vapor in contact with its liquid in a closed 
vessel, is there any limit to the pressure it can exert at a given tem- 

12. When is a vapor said to be saturated? 

13. Does the pressure that a saturated vapor in a closed vessel 
exerts depend on the volume of the vapor, on the pressure of the air 
or other substances in the vessel, or on anything but the tempera- 

14. Is a vapor in contact with its liquid in a closed vessel always 

15. When is a vapor superheated? 

16. Compare the properties of a superheated vapor with those of a 

17. When can we condense a superheated vapor to saturation by 
compression alone? When is cooling also necessary? 

18. What do we mean by critical temperature? Is it the same for 
all substances? Why is it fortunate for animal and vegetable life 
that the critical temperature of water is high while that of air is low? 

19. In what way does the large heat-absorbing power of water act 
favorably on the climate of places near large bodies of water? 

20. Under what conditions does dew settle on an object? 

•21. What is meant by relative humidity of the atmosphere? 
22. If only a small quantity of heat were to become latent heat 
when water passes into water vapor, what sort of climate would the 
earth have? 



1. The freezing temperature of water is marked 32° on the Fahren- 
heit scale, the boiling temperature 212°, and each degree on this scale 
corresponds to a temperature interval of | of a degree on the Centigrade 
scale, (a) The normal temperature of the human body is 98.4° Fah- 
renheit. How many Fahrenheit degrees is this above the freezing point 
of water? To what reading on the Centigrade scale does it corre- 
spond? To what on the absolute scale? (6) When the temperature 
of a school-room is 70° Fahrenheit, what will a Centigrade thermom- 
eter read? (c) Show that the following rule is correct: To find the 
Centigrade reading that corresponds to any Fahrenheit reading, sub- 
tract 32 from it, and multiply by J. Make up a rule for changing Cen- 
tigrade readings to Fahrenheit, (d) Mercury freezes at —38.8° C. 
What is the freezing point of mercury on the Fahrenheit scale? 

2. A student in the chemical laboratory collects 156 cm^ of oxygen 
gas at 20° C. and 78 cm barometric pressure. What would its volume 
be at 0° C. and 76 cm pressure? Its density under the latter condi- 
tions is 0.00143 151 What is its mass? 


3. How many cm' of air are there in a schoolroom whose dimensions 
are lO X 15 X 5 m? The density of air at 0° C. and 76 cm pressure 
being 0.00129 ^^, what is its mass? Its specific heat being 0.237,' how 
many calories of heat are required to raise its temperature from (.° to 
20° C? 

4. Suppose the air in the room, problem 3,. must be completely 
changed every 15 minutes, how many calories are required each hour? 
If this heat is to be taken from hot water, which enters the radiators 
at 85° and leaves them at 76°, how many gm of water must be delivered 
from the boiler each hour for this room? One gm of coal when 
burned gives up 7500 gm cal. How many Kg of coal are required per 
hour if 50 per cent of the heat of the coal gets to the radiators? 

5. When 100 gm of lead shot at 99° are mixed with 25 gm of 
water at 5°, to what temperature /will both come, if the specific heat 
of lead is 0.033? 

6. The specific heat of steam is 0.48, of ice, 0.505. Steam at 105° 
is mixed with 10 gm ice at — 10°, and the temperature is found to be 40°. 
How many calories were required to warm the ice to 0°? To melt it? 
To warm the resulting water to 40°? How much heat did each gm of 
the steam give up in cooling to 100°? In condensing to water? How 
many calories did each gram of this condensed steam give up in cooling 
to 40°? Let m represent the whole mass of the steam, and write 'the 
expression for the whole quantity of heat given up by the steam in 
changing from steam at 105° to water at 40°. Supposing no heat 
entered or left the vessel in which they were mixed, how must this 

HEAT 157 

aipaount have compared with that absorbed by the ice in warming 
to 0°, melting to water, and coming to 40°? Find the value of m, the 
mass of steam used. 

7. If 100 gm water at 50° are mixed with 200 gm ice at 0°, will all 
the ice be melted? If not, how much? If so, what will be the result- 
ing temperature? Answer the same questions if the mass of the water 
was 500 gm, and its temperature 80°. 

8. An iron girder bridge is 30 m long when its temperature is —10^ 
C. Taking the mean coefficient of expansion of iron as 0.000012, what 
is the length of the bridge when its temperature is 37° C? 

9. A bottle contains 2500 cmSof alcohol at 20° C; what will be 
the volume of this alcohol at 0° C? The coefficient of cubical expan- 
sion of alcohol is 0.0011. 


1. Repeat the experiment described in Art. 117. Have you no- 
ticed a similar phenomenon when going from a cold room to one of 
medium temperature and comparing your sensations with those of 
your schoolmates who have come from a room that is overheated? 

2. How is water purified by distillation? Petroleum consists of a 
number of different . components, each having its own boiling point; 
suggest a method for separating these components. 

3. Examine the pendulum of a regulator clock at a watchmaker's, 
and see if you can find out how the downward expansion of the pen- 
dulum rod is compensated by the upward expansion of another metal. 

4. Lay a thin strip of brass on a similar strip of iron, rivet the two 
. together, throw the combination into a fire, and see what it will do. 

Examine the balance wheel of your watch; does your observation on 
the brass-iron strip help you to explain how the watch is compensated? 

5. Examine the device (thermostat) by which the temperature of 
an incubator is kept constant. Make a diagram of it and report. 

6. How does a wheelwright put an iron tire on a wheel so that it 
will be tight? 


138. Conduction and Convection. In the preceding chapter, 
wheii heating and cooling were mentioned it was assumed that 
these terms would be understood. It may be well before wo leave 
the subject to give fixed form to our ideas concerning these 
processes. When we are heating water in a tea-kettle, we notice 
that before it can reach the water, the heat must first pass through 
the copper bottom of the kettle. Further, those portions of the 
water that are nearest the fire must become heated first. How 
do they transfer this heat to other portions of the water? Why 
does covering boiler or a steam pipe with an asbestos coating pre- 
vent waste of heat? 

In considering these questions, the first point to note is that 
when we wish to impart heat to a substance we bring it near or in 
contact with something hotter; and conversely, when we wish to cool 
it we place it near something colder. This almost instinctive prac- 
tice is based on the universally accepted concept that heat in some 
way passes from the hotter body to the cooler, and not the reverse. 
But is this strictly true? Do we never find cases in which heat 
passes from a cooler to a hotter body? Before answering these 
questions we must distinguish among several different kinds of 
heat transference. 

One form of heat transference is illustrated in the passage of 
heat through the bottom of a kettle or along a solid rod when 
one end is heated in a flame. In this case the particles of 
the kettle or the rod do not change their relative positions, 
but merely pass the heat along from particle to particle. This 
form of heat transfer is called conduction, and it is the process 
by which heat moves from one part of a solid to another. In con- 
duction, heat always flows from portions at higher temperatures 
to others at lower temperatures. 




In the case of liquids and gases, however, the process of heat 
transfer is somewhat different. All fluids are very poor conductors 
of heat, but when a small portion of the fluid becomes warmed 
by conduction from a heated body, it 
expands, and hence becomes less dense 
than the surrounding portions of the fluid. 
It is therefore pushed upward by those 
heavier surrounding portions, which 
creep in below; and as it goes it carries 
its heat with it. Thus currents are set 
up in the fluid, the cooler portions set- 
tling downward and pushing the warmer 
portions upward. 

The name of this process is con- 
vection. It is the process by which 
heat usually spreads through liquids and 
gases, and it continues as long as there 
is any difference in temperature between 
the different parts of the fluid. This 
process is illustrated in Fig. 88, as it 
takes place in a chimney. The smoke I 

shows how the hot gases flow away at Fio. 88. The Heavy Fluid 
. , . J iu i_ i_ i.i_ Displaces the Lighter 

the top, and the arrows show how the 

cold air pushes its way in below, forming a so-called "draft" along 

the floor and up the chimney. 

139. Applications of Condnction and Convection. The 

knowledge of the conducting powers of different substances is 
very useful in daily life. The asbestos coverings on locomotive 
boilers and steam pipes are poor conductors of heat, and so pre- 
vent its escape. So does wool, whether in its natural state on a 
sheep's back, or as cloth in our garments. Air conducts very little 
heat ; hence the air spaces between the walls of refrigerators. Down 
comforters are very warm for their weight, because the air, entangled 
with the down, keeps the heat from escaping. Water is a very 
bad conductor of heat; hence it must be heated from below, in order 
that convection currents may start in it. The circulation of air 


in a hot-air heating system, and of hot water in a hot-water system, 
is in many cases secured entirely by convection. The drafts up 
a lamp or factory chimney are not essentially different from trop- 
ical whirlwinds. 

140. Radiation. In both of the processes just described the 
substance heated must be in contact with the hotter body. Now 
there must be some other process of heat transfer; for we all know 
that a hot stove will heat objects in its vicinity, though not in con- 
tact with them, to a temperature higher than that of the surround- 
ing air, and that the life-giving energy of the sun somehow suc- 
ceeds in reaching the earth without any apparent contact between 
the two. Hence, for describing this process of heat transfer, we 
are compelled to imagine another mechanism which is called 


141. Diffusion. We can most easily gain some conception 
as to how radiation takes place, if we pause for a moment 
to consider some other phenomena which will enable us to form 
an idea as to what heat is. When any gas having an easily recog- 
nizable odor, such as illuminating gas, is liberated in a room, 
in a very short time the odor can be detected in every part of the 
room. This familiar fact justifies us in concluding that the gas 
has spread throughout the entire space. How may this have hap- 
pened? Evidently the gas must consist of numerous particles, and 
these particles must have moved from the place where they were 
liberated into all other portions of the room. Hence we are led 
to think that the particles of this substance must have been in 
motion before they were liberated, and that to liberate them it was 
merely necessary to open a stopcock, or take out a cork, so as to 
provide an opening in the confining walls, through which the gas 
particles might escape. The simplest idea that we can form of 
their motion is that each particle is highly elastic and that it con- 
tinues to move in a straight line until it collides with some other 
particle or with the sides of the vessel; when it immediately re- 
bounds and starts off again in a new straight path. 


142. Evaporation. Let us now consider whether this 
hypothesis will help us to a better understanding of the phenomena 
of evaporation, of which we learned in the preceding chapter. 
We there found that water evaporates at all temperatures, i;e., the 
water particles fly away from the liquid surface and diffuse them- 
selves into the surrounding air. If we assume that the water 
particles are in violent motion while within the body of the liquid, 
we can form a mental picture of how the particles that are at the 
surface might be more free to fly out into the air than to fly back 
into thQ liquid. Further, since a vapor always occupies so much 
more space than does the liquid from which it has been formed, 
we must conclude that the particles of water vapor are farther 
apart than are those of liquid water, and therefore we can under- 
stand why they have far greater freedom of motion. 

143. Diffusion of Solids. Again, we are familiar with what 
often happens when we put a solid into a liquid. For example, a 
lump of sugar, placed in a cup of coffee, disappears after a time, even 
without stirring; and, if left long enough, it will sweeten all 
the coffee in the cup. Here again we are led to conclude that the 
sugar and the liquid consist each of a large number of particles which 
are already in motion, and that when the solid is put into the liquid, 
the moving particles of each spread themselves amongst those 
of the other. When the particles of two or more substances thus 
spontaneously mix together, they are said to diffuse into one another, 
and the process is called diffusion. 

144. Gaseous Pressure. With the aid of our hypothesis 
we can now get a very satisfactory conception of the manner in 
which a body of gas exerts a jpressure on the walls of a vessel in 
Vhich it is confined. For it is easy to see that if the millions of tiny 
gaseous particles are flying with great velocities in all directions, 
the sides of the vessel will be struck at every instant by a multitude 
of these particles, and that each of these when it strikes and 
rebounds will give the wall a push or impulse. The magnitude 
of this impulse will depend on the mass and velocity of the particle 
(c/. Art. 39). The sides of the vessel will thus be bombarded 

162 . PHYSICS 

at every instant by a multitude of the little particles, which come 
so thick and so fast that the sum of their impulses can not be dis- 
tinguished from a continuous pressure. 

145. Effect of Heating. We have learned, (Art. 121) that 
when a gas is heated in a closed space at constant volume, the 
pressure that it exerts is increased; but we have just seen that the 
pressure depends both on the masses and on the velocities of the 
gaseous particles. Hence, since heating the gas does not change 
the mass of the flying particles, it must increase their velocities. 
Also, since at a given instant each particle has a certain mass ?m, 

and a certain velocity v, it has a certain amount of kinetic energy — ; 

and since, as we have just found, heating increases the velocity 
t;, we are forced to conclude that what the heat energy does when 
it raises the temperature is simply to increase the kinetic energy 
of the little particles. Thus we are led to infer that heal is nothing 
more nor less than the kinetic energy of these moving particles, 

146. The Kinetic Hjrpothesis. The hypothesis at which 
we have now arrived includes the following ideas: 

1. Every substance consists of a great number of very small par- 
ticles, each of a definite mass. These particles are called mole- 

2. These molecules are constantly in rapid motion. 

3. Heat is the kinetic energy of these moving molecules, 

4. The temperature of a body depends on the average kinetic 
energy of the individual molecules of the mass, while the total quxm- 
tity of heat possessed by it depends on the sum of the kinetic energies 
of all its molecules. 

This kinetic hypothesis has helped us to a better understanding 
of diffusion, evaporation, and gaseous pressure; let us now return 
to radiation and see if it will assist us there. 

147. Radiation. We have learned that radiation consists in 
the transfer of energy from one body to another when the two are 
not in contact, as, for instance, from a stove to your hand. Now, 


if the particles of the stove are in rapid motion, and if heating 
the hand consists in making its particles move more rapidly, 
by what possible mechanism may t6e rapidly moving particles 
of hot iron communicate some of their motion to the particles 
of your hand across the intervening space? Does anything like 
this happen when a stone is thrown into a pool? Does not the 

Fig. 89. Energy May be Transmitted by Waves 

motion of the stone produce waves, as sho^vn in the photograph 
(Fig. 89), which move in gradually widening circles until they 
reach the borders of the pool? What happens to the pebbles there 
when the waves reach them, and what becomes of the waves 
themselves? Do they not set the pebbles in motion, thus passing 
along to the pebbles the energy that they received from the stone? 
May we not, then, imagine that just as water waves spread out in 
rings from the spot where a stone falls, so heating waves spread out 
in all directions from the vibrating molecules of hot bodies; and 
just as water waves break and give up their energy to pebbles on 
the shore of the pond, so the heating waves strike and give up 
their energy to particles on the boundaries of their realm? 

148. The Ether. In the case of the pebbles that are set 
in motion when a stone is thrown into a pool, it is evident that 
water is the medium through which the energy travels, and that 
without some such medium, no energy can be thus transferred. 
Now if heat is transferred by waves, what is the medium in which 


these waves are propagated? That it is not air, nor ordinary 
matter of any kind, must be manifest to any one who will but hold 
his hand near an electric glow lamp; for the air has been pumped 
out of the bulb, yet the filament radiates both heat and light through 
this vacuum. So also we are warmed by the energy that comes 
to us from the sun, although there are good reasons for believing 
that the greater part of the space between does not contain any sen- 
sible amount of ordinary matter. Hence the adoption of the wave 
hypothesis for radiation makes it necessary to assume that there 
exists in space a medium which is not ordinary matter, but which is 
capable of transmitting such waves. This medium is called the 
ETHER. When we come to study electricity and light, we shall 
meet with it again, and, are likely to become more deeply im- 
pressed with the utility of the ether-wave hypothesis. 

149. Prevost's Theory of Exchanges. Suppose that the 
fire goes out so that the temperature of the stove falls to that 
of the hand; then you can no longer warm it at the stove. Has 
the stove then ceased to send out heat waves? If now we bring 
a piece of ice near the stove, will not some of it melt? Yet when 
the stove has further cooled to the temperature of the ice, this latter 
will no longer be melted by the heat from the stove. Has the stove 
then ceased to send out heat waves? Is it not simpler to suppose 
that all bodies are radiating heat waves at all temperatures, and 
that whether a body grows hotter or colder depends on whether, 
in a given time, it absorbs more than it radiates, or radiates more 
than it absorbs? 

We may state this theory as follows: All bodies are radiating 
and absorbing heat energy at all times. If, in a given time, the 
amount of energy that a body absorbs is greater than that which 
it radiates, the temperature of tJie body rises; while if the amount 
of energy that it absorbs is less than that which it radiates, its tem- 
perature falls. This theory was first propounded by Prevost, and 
hence it is known as Prevost's Theory of Exchanges. 

160. Absorption. Radiant heat, though not absorbed by 
the ether, is always absorbed to a greater or less degree by ordi- 


nary matter. Elaborate experiments have been made to determine 
how much of the energy of radiant heat is absorbed by various 
substances; and it has been found that of all the gaseous 
substances, water vapor has the largest absorbing power. Thus 
the experiments of John Tyndall showed that a layer of air satura- 
ted with water vapor at ordinary temperatures, and four feet thick, 
absorbs 20% of the radiant heat energy that falls upon it. Of 
course solids and liquids absorb more of the radiant heat than this. 

151. Absorbing Power of Water Vapor. The fact that water 
vapor is a powerful absorber of radiant heat furnishes us with 
another admirable example of nature's adaptation of means to 
an end; for what would be the condition of the earth's surface 

Fig. 90. Glacier and Snow-Field on a High Mountain 

if the water vapor did not absorb a large portion of the sun's 
energy? We should be exposed to a blistering heat in the day- 
time, and a freezing temperature at night. That this is actu- 
ally the case on high mountains, where the earth is not so thickly 
blanketed with water vapor, is well known to every one. This 
is largely due to the fact that water vapor is transparent to 
light, i.e., it absorbs very little of the sun's energy that reaches 
it in that form; while on the other hand it absorbs a very large 
proportion of the radiant heat waves. When the light energy 
from the sun has passed through the vapor-laden atmosphere, it 
is absorbed by the bodies on the surface of the earth, and is there 
converted into heat. At night, when these bodies arfe sending out 
this energy as radiant heat, and are radiating more energy than 
they are receiving, the water vapor in the atmosphere absorbs 



this radiant heat, and prevents it from escaping into space. Thus, 

acting like a trap to catch and hold the sunbeams, the water vapor 

accumulates heat energy in the day-time and 

throughout the summer, and holds it over for the 

nights and the winter. 

The glass of a hothouse acts in a similar 
way: it lets the light in; but when the light has 
been absorbed and converted into heat, the glass 
will not let it escape by radiation. So it remains 
and keeps the plants warm. For this reason 
vegetables can be grown under glass very early 
in the spring-time without the aid of artificial heat. 

Fig. 91. Automo- 
bile Cooler 

152. Radiation and Absorption. A close relation is found to 
exist between radiation and absorption. Substances that send out 
large amounts of radiant heat and light when red hot, are found 
also to absorb large quantities when cold. Thus, a substance like 
lampblack is dark when you look at it because it absorbs nearly 
all the light that falls on it; 
it is also found to be a pow- 
erful absorber of radiant 
heat. We all know well that 
this substance is a splendid 
radiator, for carbon is used 
in the manufacture of elec- 
tric lights, and is present in 
large quantities in oil and 
gas flames. Furthermore, 
coal and wood-charcoal, 
which are also forms of car- 
bon, when burning become 
powerful radiators of both 
heat and light. 

Since radiation and ab- 
sorption take place only at the surfaces of bodies, it follows other 
things being equal, that the greater the surface the greater is the 
radiation or absorption. Hence radiators for heating houses and 

Fig. 92. Detail Fig. 91 Showing Air Cells 

ING Surface 



for cooling automobile engines are better if black and rough than 
if bright and polished, because then they radiate more outward and 
reflect less inward for a given surface, and because when they are 
rough they have a larger surface. 

163. Heat and Light. When any substance is heated, as in a 
blacksmith's forge, it becomes red-hot, at a temperature of 520° C, 
i.e., it begins to give out red 
light; and if it is further 
heated, it not only gives out 
more heat, but the light that 
it emits becomes first yellow 
and then white. Since 
bodies at high temperatures 
give out both radiant heat 
and light, and since both ra- 
diant heat and light are con- 
verted into heat when they 
are absorbed, it appears that 
they must be closely related 
phenomena. It is, therefore, 
reasonable to suppose that if 
the radiant heat is a wave 
motion, light is also a wave 
motion. If this conjecture is 
correct, how does light differ 
from radiant heat? 

We may imagine that, just as on a calm day we have the 
long, steady roll of the ocean, so from bodies at low temperatures 
we have long heat waves; and on the other hand, just as, during 
a storm, we have not only that same long roll of the ocean much 
intensified, but also a multitude of shorter waves added to it, so 
from red-hot bodies we have not only the long radiant heat waves 
intensified, but also shorter waves which give us the sensation of 
light. We shall have occasion to test tliis assumption in a num- 
ber of ways when we come to take up the detailed study of light 
in the last five chapters of this book. 

Fig. 93. The Hot Iron Sends Out Waves 
OF Various Lengths 



1. Heat transference is of three kinds, conduction, convection, 
and radiation. 

2. The facts of diffusion, evaporation, and gaseous pressure 
lead to the hypothesis that heat is molecular kinetic energy. 

3. The facts of radiation suggest the hypothesis that radiant 

Fig. 94. The Big Roller Carries Smaller Waves on its Back 

heat and light are forms of wave motion in a medium called the 

4. All substances are radiating heat waves at all temperatures. 

5. A body does not change its temperature when it receives 
in a given time as much radiant energy as it itself sends out. 

6. The amount of radiation depends: 1, on the radiating 
substance; 2, on the difference in temperature between the radiating 
body and the surrounding space; 3, on the amount of surface 


1. How do we describe the process by which heat travels along 
an iron rod when one end of it is held in a flame? What name do we 
give to the process? 

2. What can you say of the conductivities of different substances, 
solid, liquid, and gaseous? 

3. What is the name of the process of heat transfer in fluids? De- 
scribe the process, and show how it differs from conduction. 

4. When bodies are not in contact, how do we conceive that heaJb 
travels from one to the other? What is the name of this process? 

5. When two bodies not in contact have the same temperature, 
what must be their action with regard to radiation? 

6. Why do we suppose that all bodies are sending out radiant 
heat waves at all temperatures? 


7. Do all bodies at the same temperature have the same radiating 
power? What ones radiate most? 

8. What relation has the radiating power of a substance to its 
absorbing power? 

9. What are some of the conditions that are favorable to rapid 

10. Mention some practical applications of conduction, convection, 
and radiation. 


1. Why are wooden handles put on short fire pokers, but not on 
long ones? 

2. At what level should the cool water return pipe and the cold 
water supply pipe enter the hot water reservoir of a water heating 
system operated by convection? From what level should the hot 
water main leave? Should the heating pipe in the stove or furnace 
be horizontal or inclined? Which should be at the higher end, the 
cold water supply pipe or the hot water delivery pipe? Give reasons. 

3. How are both evaporation and diffusion illustrated by liquid 
perfumes? Do solid perfumes act in the same way? 

4. Do clothes dry while frozen, when they are himg out on the line 
in freezing weather? Does snow ever disappear from the ground 
without melting? Do crystals ever form on the sides of a bottle of 
gum camphor? Do solids never pass from the solid to the vapor state 
without first passing into liquid? 

5. Do you become warmer in summer if you wear a black suit than 
if you wear a white one of the same material? Why? Why is loosely 
woven, thin material more comfortable to wear in summer than thick, 
compact material? 

6. What is the use of tarred paper in the walls of houses? In what 
way do double windows save heat? 

7. By what process or processes is heat distributed, when a room 
is heated by hot water radiators? By steam radiators? By indirect 
radiators? By hot air registers? By stoves? By grates? Can you 
get hot air to come out of a register into a room that is tightly closed? 

8. What advantage arises in the hot water heating system from the 
high specific hect of water? In the steam system, from the high latent 
heat of steam? 

9. Why does a room become so very hot in summer when the sun 
shines into it through a closed window? 

10. Sketch an arrangement for cooling a house by means of a fur- 
nace which is to consume ice instead of coal, and send cold air instead 
of hot air out of the registers. Sketch a similar scheme for reversing 


the action of a hot water heating system and an ordinary stovcf, speci- 
fying in each case the locf^tion of the cooler in the house. 

11. Why is the word *' draft" an inappropriate one to use in connec- 
tion with convection? 


1. With a thermometer, take the temperature of carpet, oil-cloth, 
metals, and wood in a cold room. Feel each of the substances with 
the hand. Explain why the sensations do not agree with the indica- 
tions of the thermometer. Make the same experiment with several 
substances after warming them in an oven. 

2. Stir some scrapings of blotting paper into a glass of water, add 
a lump of ice, and make a diagram of the convection currents that are 
shown by the scrapings. Make the same experiment, using a hot clay 
marble instead of the ice. 

3. Examine a refrigerator. Do you find any provision for securing 
a circulation of air in it by convection? If so give diagram and brief 

4. Investigate a hot air heating system, and hand in a diagram 
with a brief explanation of how the air circulates, and how it heats the 
rooms. Do the same for a hot water system and a steam system 
Visit the stores where such heaters are on sale, and ask for information 
illustrated circulars, etc. Investigate particularly the heating system 
in your own home, and give in a brief paper the results of your ex- 
perience as to the best methods of managing it, especially the regula- 
tion of the cold air supply. 

5. Read the chapter on Heat in Experimental Science^ by Geo. 
M. Hopkins (Munn & Co., N. Y.). Many interesting and easy 
experiments are described in it. 

6. Tyndall's Heat as a Mode of Motion (Appleton, N. Y.) 
contains the descriptions of many interesting experiments in conduc- 
tion, convection, radiation, and absorption. Much of the theory of 
this book is, not up-to-date, but the facts^ told in TyndalPs charming 
and masterly style, are well worth reading. Some of the experiments 
may easily be repeated. 


154. The Mechanical Equivalent of Heat. Let us now pass 
finally to the consideration of the relations between heat and 
mechanical work. Nearly everybody has polished a cent by rub- 
bing it on a carpet, or seen sparks fly from a grindstone when a 
tool is being ground, and therefore knows that heat can be pro- 
duced by mechanical work. No definite relation between work 
and heat was recognized in the early days of science. When Sir 
Humphry Davy, in 1810, showed that he could melt two pieces 
of ice simply by rubbing them together, the idea that heat could 
be produced by mechanical work began to prevail. 

Since heat can be converted into work, the question at once 
arises, how much work can one gram calorie of heat do? The 
first to solve this problem experimentally was James Prescott 
Joule (1818-1889), who attained the result in a very interesting 
and instructive way. His apparatus is pictured in Fig. 95, and 
consisted of a calorimeter filled with water, in which paddles were 
made to rotate by the weight of a falling mass. Thus the energy 
of the falling mass is expended in heating the water by friction. 
The amount of the work done is measured by the product of the 
weight of the mass and the distance through which it falls. The 
number of calories of heatv generated is the numerical product of 
the specific heat of the water, its mass, and the change of tem- 
perature {cf. Art. 126). More recently Rowland, at Johns Hop- 
kins University, made a most careful and accurate determination 
of this constant. The method employed by him was not different 
in principle from Joule's. The result of his experiments has been 
adopted as the jnost probable value of the number of ergs that 
is equal to one gram calorie. It is 

1 gm cal = 4.19 X 10^ ergs. (9) 

This ratio is known as joule's equivalent, or the mechanical 





155. A Gas is Heated When it is Compressed. Who has not 
noticed that a bicycle pump becomes heated when it is being used 
to pump up tires? Since a great deal more heat is thus developed 
than can be attributed to friction, we are forced to conclude that 

the greater part of it is produced by 
the work done in compressing the 
air. This is perhaps the most 
direct and simple case of the trans- 
formation of energy into heat. 

®With the help of the kinetic 
hypothesis, which we adopted in 
the preceding chapter, we may eas- 
ily form a mental picture of the way 
in which the heat is added to the 
gas when it is compressed. For if 
the piston of the pump is moving 
downward, while millions of the 
little molecules are flying upward 
against it, each little molecule Will 
rebound with an increased veloc- 
ity, just as a base ball rebounds 
from a moving bat. But this in- 
creased velocity implies an increase 
in the kinetic energy of the mole- 
cules; and the total kinetic energy 
that has been thus added to all the 
molecules — i.e., the added heat — is the equivalent of the work 
done by the advancing piston in compressing the gas. 

Fig. 95. Joule's Apparatus 

a. Revolving Blades. 

b. Stationary Blades. 

166. A Gas Cools When it Expands. That a gas cools when 
it expands also follows at once from our hypothesis. For just as a 
base ball strikes against the hands of the catcher and gives up its 
kinetic energy in putting them into motion, so the little molecules, 
when they bombard the piston so as to put it into motion, must 
give up some of their kinetic energy in doing this work. The total 
kinetic energy thus lost by the molecules — i.e., the lost heat — is 
the equivalent of the work done by the gas on the piston. 


It will make no difference whether the piston is present or 
not; for if a compressed gas is allowed to expand directly into the 
air, it must do work in pushing aside the air; and it must lose 
heat in order to do this work. Doubtless many of you have made 
an experiment that verifies this prediction; for you may have held 
your hand near the valve of your bicycle tire when the compressed 
air was escaping from it, and observed that the Jet of air seemed 
very cold. You may also have noticed that when a bottle of 
ginger ale or pop is opened, a cloud of condensed vapor appears 
near its mouth. The gas escaping from the bottle becomes so 
cold from doing the work of pushing away the air, that it cools 
this air below the dew point. . 

157. Liquid Air. This principle is the one that is used in 
the liquefaction of gases, notably of air. The air is first cooled 
and compressed as far as possible, and is then allowed to escape 
through a small opening into the room. In this escape the at- 
mosphere must be pushed back by the expanding air; and heat must, 
therefore, be supplied to do this work. The only heat immediately 
available is that still remaining in the compressed and cooled air. 
So much heat is taken from it that part of it is condensed into a 

168. Cooling by Evaporation. In Chapter VII we learned 
that a large 3,mount of heat must be supplied in order to change 
a liquid into a vapor. We are now in a position to appreciate 
that this heat is employed in doing the work of changing the 
liquid into the vapor. We can also understand that if this heat 
is not supplied from some external source, it may be obtained 
from the liquid itself, in which case the temperature of the liquid 
falls. When we recall the fact that the amount of heat required 
to vaporize liquids is usually very large — ^in the case of water 
for example, more than 500 gm calories per gm — we can see why 
this principle may be advantageously employed in cooling proc- 
esses. By the rapid evaporation of liquid air, temperatures 
lower than —182° C. may be obtained. 


159. Manufactured Ice. Ice is now made in all large cities, 
and in the process the principle just discussed is applied. Am- 
monia or carbon dioxide gas is condensed into a liquid by means 
of a powerful force pump driven by a steam engine or an electric 
motor. The liquefied gas is then allowed to escape through a 
valve into a system of pipes from which the air has been pumped. 
These pipes pass back and forth in a large tank which is filled 
\^ith salt water, so that the heat required for evaporating the 
liquid carbon dioxide, is taken from this brine. This is then 
cooled until its temperature is several degrees below 0° C; but 
it does not freeze, because the freezing point of salt water is lower 
than this. The pure water that is to be frozen is placed in large 
iron molds, which are submerged in the cold salt water, and are 
kept there until the water in them is frozen. 

160. Cold Storage. Cold storage rooms are generally op- 
erated in connection with artificial ice plants. These rooms have 
thick walls made of nonconducting materials, and around them 
on the inside are rows of pipes. The brine from the freezing 
tank^ is pumped through these pipes on its way back to the cool- 
ing tank, and thus serves to reduce the temperature of the room 
to the point desired. Immense quantities of eggs, butter, fruit, 
and other perishable foods are thus preserved in cold storage for 
use in the winter months. 

161. The Steam Engine. We are now prepared to consider 
the way in which heat is converted into mechanical work by 
steam engines. We select the locomotive as a typical case of 
these 'engines, because it is complete in itself. Fig. 96 shows 
the construction of a modern locomotive. The heat is derived 
from a fire, which is built in the fire-box Fb at the rear end of the 
boiler. In order that this heat may pass easily into the water in 
the boiler, the smoke and hot gases produced by the fire are sent 
through a large number of tubes Ft, which pass through the boiler 
from the fire-box to the forward end. The water in the boiler 
surrounds these tubes and the fire-box, so that it is in a position 
to absorb a large part of the heat of the fire. When the engine 



QQ H »^ •^ 

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5? H ^ ►tj 
<l> 3* f1" =• 

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2.^ cr Q 

2. s; ^ a. 
o 3 2 » 




•a o 

5 '^ w w 

- ^ ? " 

P c^ p c 
o o ^ ^ 



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has "steam up/' the upper part of the boiler is filled with steam 
at high pressure. 

When the engineer wishes to start the engine, he pulls on a 
lever called the throttle; and this lever opens the throttle valve 

Tvy which is placed at the highest 
point in the boiler. When this 
valve is opened, the steam rushes 
into the large supply pipe, which 
conducts it to the steam chest Sc, 
whence it passes to the cylinder C. 
^'°- ^^* rSRw^ARr'' ^''^^''^ Having reached the cylinder through 

the port p, the steam pushes against 
the piston P and moves it backward. This motion of the piston 
is transmitted by means of the piston rod Pr and the connecting 
rod Cr to the driving wheels Dw, which are thus made to turn. 
When the piston has reached the rear end of the cylinder, the 
slide-valve automatically opens the port p', and also connects the 
port p with the exhaust e, Fig. 97. The high pressure steam in the 
steam chest then rushes into the rear end of the cylinder and 
pushes the piston forward, driving the steam in the forward end 
out through the port p and the exhaust e into the exhaust pipe 
Ep, whence it escapes with a puff up the stack. The valve then 
automatically connects p with the steam chest and p' with the 
exhaust and the piston is again pushed backward, as in Fig. 98; 
and so on. Thus the steam, by expanding in the cylinder, moves 
the piston; and this motion is used 
in doing the nTechanical work of 
turning the drivers. 

162. Work Done by the Steam. 
The most important question about 
a steam engine is: What is its Fio. 98. The^ piston Starts 
efficiency? In order to answer 

this question, we have to determine how much work is done by 
the steam, and how much energy is supplied to it (cf. Art. 37). 
How can we do this? How shall we measure the work done by 
the steam? How determine the amount of heat energy supplied? 


The work done by the steam may be determined with the help 
of equation (5), Art. 34, W = fl. For if the total force of the 
steam against the piston is /, and if this force acts through a dis- 
tance / equal to the length of the stroke, the work done at each 
stroke of the piston is simply the product of these two quanti- 
ties. But the total force of the steam on the piston is the 
steam pressure, i.e., the force per cm^, multiplied by the area 
in cm^ of the piston. Therefore the work done by the steam is 
W = steam pressure X piston area X length of stroke. 

This result may be interpreted in another way. For piston 
area X length of stroke is the volume of the cylinder; therefore, 
the work done in one stroke is W = steam pressure X volume of 
cylinder. Hence we can determine the work done by the steam 
when we know these two factors. 

In actual practice it is an easy matter to measure the volume 
of the cylinder; but the pressure is not constant throughout the 
stroke. When the steam enters the cylinder, it has the pressure 
that exists in the boiler; and, if the port were left open during the 
entire stroke, the steam would be exerting that same pressure 
when the exhaust was opened. It would then expand suddenly 
into the air, and thus waste a large amount of its energy. There- 
fore it is customary to arrange the, valve V, Fig. 96, so that 
it closes the port p when the piston has made about one-quarter 
of its stroke. The steam then pushes the piston the remaining 
three-quarters of the stroke by its own expansion. But while the 
steam is expanding and doing this work, its pressure is decreas- 
ing; therefore when the exhaust is opened, the steam pressure 
is found to have fallen nearly to that of the atmosphere. 

163. The Pressure- Volume Graph. Since the pressure of 
the steam changes during the stroke, we must find an average 
value to use in calculating the work. This is generally done by 
measuring the pressure at different parts of the stroke by means 
of an ingenious pressure gauge attached to the cylinder (c/. 
problem 8, page 187). The pressures thus found are plotted as 
ordinates with the corresponding volumes as abscissas, and the 
average pressure obtained by measurements on the graph. Fig. 


99 is such a graph taken on a locomotive cylinder when it was 

pulling a train. The point p^ represents the pres;sure and v^ the 

volume when the steam had just entered the cylinder. As the 

piston moved, the volume t\ increased, while the pressure remained 

constant and equal to 

fiQ, C^ that in the boiler until 

the point pj was 

reached. At this point 

the port was closed and 

the steam, being thus 

"cut off" from the 

boiler, was left to ex- 

^^ ^^ pand. During the re- 

Fio. 99. The Pressure-Volume Graph . , « .| . , 

mamder of the stroke, 
while the steam was expanding and the volume increasing, the 
pressure was falling as shown by the curve p2 p^. At p^, the 
piston having now reached the end of its stroke, the exhaust 
was opened, so that the pressure fell suddenly to that of the 
atmosphere, represeifted by p^. As the piston returned, the 
volume of the cylinder decreased; but because the exhaust was 
open the pressure there remained constantly equal to atmos- 
pheric pressure up to the point p^. At this point the valve 
shifted so as to close the e^chaust and let in the live steam (c/. 
Art. 161); and consequently the pressure jumped quickly to p^, 
that of the boiler, and the entire process was repeated. 

164. Back Pressure. In determining the average pressure 
from a graph like this, one thing must be carefully noted. When 
the piston returns, it has to push the exhaust steam out of the 
cylinder against the pressure of the atmosphere. The work 
of doing this is numerically equal to atmospheric pressure X 
volume of cylinder. This amount of useless work must be sub- 
tracted from that done by the steam, in order to find the amount 
of available mechanical work done by it. Therefore, when an 
engine exhausts into the atmosphere, as the locomotive does, 
the average pressure by which its work is determined is the differ- 
ence between the total average pressure and that of the atmosphere. 


165. Lower Pressure at Exhaust. It has probably occurred 
to many of you that we might increase the amount of work done 
if we could allow the gteam to exhaust into a vacuum instead of 
into the air; for then no work would have to be wasted in pushing 
the used up steam out of the cylinder. This may be partially 
accomplished by allowing the exhaust steam to escape into a coil 
of pipes which contain no air, and which are surrounded by cold 
water. When the exhaust steam enters these pipes, it is con- 
densed into water, and so the pressure there is that of the saturated 
vapor at the temperature of the cold water that surrounds the 
pipes (cf. Art. 128 and Fig. 85). Such a device is called a 
CONDENSER. The water condensed from the steam has to be 
pumped out of the condenser against atmospheric pressure; but 
since its volume is so much less than that of the steam, the useless 
work done against the atmospheric pressure is much less. Con- 
densers can not be used on a locomotive, because they are too 
bulky, and because they require a large amount of cold water 
to keep them cool. They are useful, however, on steamboats, 
where space is more plentiful, and where a large supply of cool 
water is always at hand. 

166. Increased Boiler Pressure. Another way to increase 
the amount of work done by the steam is to increase the boiler 
pressure by increasing the temperature of the steam ; for the work 
= average pressure X volume of cylinder, and this process in- 
creases the average pressure. There is, however, a practical 
liniit to increasing the efficiency in this way, because the pressure 
must never be allow^ed to approach that at which the boiler would 
burst {cf. Arts. 128, 129). 

167v Heat Energy Consumed. Having now found how the 
work done by the steam may be measured, let us consider how 
the quantity of heat energy supplied to the engine is determined. 
This is done practically by first noting the amount of fuel con- 
sumed by the engine in a given time, and then finding out how 
many heat units are liberated when this amount of fuel is burned. 
For example, a locomotive of the type shown in Plate I con- 


sumes 1500 gm of coal for every horse-power that it furnishes 
for one hour. The number of gm cal liberated by each gm of 
coal when it is burned is found to be 7800. Hence the amount 
of heat liberated by 1500 gm of coal is 1500 X 7800 = 117 X 10^ 
gm cal. Since the mechanical equivalent of 1 gm cal is 4.19 X 
10^ ergs [c/. equation (9), Art. 154], this amount of heat is equal 
to 117 X 10^ X 4,19 X 10^ = 49 X 10*' ergs. This is the energy 
supplied to the engine for each horse-power hour. 

Since (cf. Art. 43) 1 horse-power = 746 X 10^ — ^, and since 
' - ^ sec - 

1 hour = 3600 sec, 1 horse-power hour = 3600 X. 746 X 10^ 

= 268 X 10" ergs. This is the work done. The efficiency 

is then ^ork done 268 X 10" - ^^ - q 055 - 5 5^^ 

"" ^^^""^ energy supplied ~ 49 X lO^''^ " 4900~ ^•"^^'" ^'^^'' 

168. Efficiency and Temperature. We have just found 
that a locomotive, when considered from the point of view of 
efficiency, is a very inefficient machine. Yet even stationary 
and marine engines seldom have efficiencies greater than 17%. 
This fact leads us to ask whether there is any theoretical 
reason for this. Are the conditions under which every heat 
engine must work such that its efficiency is necessarily small? 
Or may we hope some day to make heat engines of high 

We can find answers to these questions by carefully tracing 
the heat through an engine. We note that heat is absorbed by 
the water when it passes into steam. The steam then carries 
this heat with it when it goes into the cylinder, where part of the 
heat is used up in doing the work of moving the piston. The 
rest of the heat is carried with the exhaust steam into the air or 
condenser, and is then no longer available for doing work in the 
engine. The essential things in this process are: 1. Heat en- 
ergy is imparted to the steam at a high temperature (boiler tem- 
perature); 2. The temperature of the steam must fall when work 
is done; 3. Heat is given up to the condenser at a low temperature. 

The only heat available for doing work is that given up by the 
steam in cooling from the temperature of the boiler to that of the 

Plate VI. A Triplk Expansion Pumping Engine, 
Baden Station, St. Louis 


condenser. Hence the heat energy available for work depends 
on this difference in temperature. An engine that could convert 
all of this available heat energy into work would be a perfect 

The total heat energy in the steam when it leaves the boiler 
depends in a similar way on how many degrees the boiler tem- 
perature is above the absolute zero. We may then say that the 
total amount of heat energy of the steam depends on its absolute 
'temperature, i.e., on 273 + t (cf. Art. 123). 

Finally, since the efficiency of an engine is defined as 

heat converted into work ' . . . . . ., , 

— 7-r-n — I r-i — » and smce m a perfect engme the work 

total heat supphed ^ ® 

depends on the difference in temperature (t — t') between the 

boiler and the condenser; and since the total heat energy sup-^ 

plied depends on 273° + t, we may conclude that the efficiency of 

a perfect engine might be expressed by the ratio -^=^ — -. 

169. Comparison of Efficiencies. Plate VI is a picture of 
one of the large pumping engines that supply the city of St. Louis 
with water. It is a triple expansion engine. The steam from 
the boiler enters the smaller cylinder at the left of the picture and 
there expands and cools somewhat. It then passes into the 
second cylinder, in which it expands and cools some more; and 
from this it exhausts into the third and largest cylinder, at the 
right of the picture. From this it passes to the condenser. This 
arrangement, by which the steam is allowed to do part of its 
expanding in each of the three cylinders, has many practical 
advantages, such as distributing the strains among three cranks 
instead of concentrating them in one; greater compactness, 
since a single cylinder that would do the work of the three would 
have to be of enormous size; greater economy in steam, since the 
fall in temperature in each cylinder is only one-third of the total 
fall, so that the steam does not condense so readily into water in 
the cylinder. 

In this engine the steam has a pressure of 10 atmospheres 
(760 cm of mercury) when it enters the first cylinder, and it ex- 


hausts into a condenser in which the pressure is 53 cm of mercury. 
On consulting a table in which the relations between temperature 
and pressure of saturated water vapor are given, we find that the 
temperature corresponding to the boiler pressure is 180° C, and 
that corresponding to the condenser pressure is 80° C. If 
the engine were perfect, its efficiency would then be 

<-r 180- 80 100^ ^ 
273+^273+180 453 '''' 

It was found that 500 gm of coal was consumed at the boiler 
for every horse-power hour furnished by the engine. Since the 
locomotive just considered (Art. 167) consumed 1500 gm of coal 
per horse-power hour, the efficiency of this engine is just three 
times that of the locomotive, or 5.5 X 3 = 16.5%. The real 
engine is. thus seen to be about three-quarters perfect. 

Sometimes boiler pressures as high as 15 atmospheres are 
used. The corresponding temperature at the boiler is found 
from the table to be about 200° C. If the pressure in the con- 
denser is reduced to 3 cm of mercury, the corresponding tempera- 
ture would be 30° C. So the efficiency of a perfect.engine working be- 

i~i' 200— 30 170 
tween these temperatures would be 273^^^ = ^73 + 200 " 473 " ^^^^ 

nearly. Since boiler pressures greater than 15 atmospheres 
are not very safe, and since it is very expensive to reduce the 
condenser temperature below 30° C, 36% represents the prac- 
tical limit for the efficiency of a perfect steam engine. No 
steam engine has yet been made with an efficiency as high as 

170. The Gas Engine. The efficiency of a steam engine is 
thus seen to be low because the range of temperature (t—f) 
through which we can use the steam is comparatively small. In 
the gas engine, Fig. 100, the conditions are more favorable. In 
this machine the fuel is burned in the cylinder where the work is 
done. The temperature of the gas in the cylinder may, therefore, 
be very high. * A mixture of gas and air is introduced into the 
cylinder and there exploded. The pressure developed by this 
explosion pushes the piston outward. It is then pushed back 



Fig. 100. A Gas Engine 

by the atmospheric pressure: another explosion pushes it outward 
again, and so on. 

A good gas engine consumes about 16 cubic feet of gas from 
the city mains for every 
horse-power hour that it sup- 
plies. The heat of combus- 
tion of illuminating gas in 
New York has been found to 
be 18 X 10* gm cal per 
cubic foot. Hence the heat 
supplied by 16 feet of gas 
is 16 X 18 X 10* = 288 X 10* 
gm cal. This heat energy 
is equal to 288 X 10' 
X 4.19 X 10' ■■= 1200 X 10^' 
ergs. This is the heat en- 
ergy supplied to the en- 

In Art. 167 the number 

of ergs in a horse-power 

U^,,« „, ^ f^„.wl 4^ U^ Fig. 101. The Old-Fashioned 

hour was found to be Water -Wheel 



268 X 10^^ ergs. Therefore the efficiency of this gas engine is 
268X10" 22%. 

1200 X 10^' 

171. Turbines. In the engines thus far considered the rotary 
motion of the drivers or of the flywheel was produced by a trans- 
latory motion of the piston to and fro. Engines of this type 
are therefore called reciprocating engines. In all such engines, 

Fig. 102. A Steam Turbine 

useless work has to be done in starting and stopping the pis- 
ton at each stroke; and this action alw^ays produces a jarring 
which is harmful both to the engine and to the building or boat 
in which it is placed. 

The old-fashioned water-wheel, Fig. 101, and the modern water 
turbines, such as are used so extensively at Niagara, illustrate 
another method of converting kinetic energy of translation into 
kinetic energy of rotation. The moving water is projected 
against the paddles or blades of the wheel, and thus keeps it 
steadily turning. Many attempts have been made to construct 



an engine in which a wheel would be set into rotation by blowing 
steam against blades or paddles on it. It is only within the 
last few years that engineers have learned how to apply this 
principle so as to make a steam' turbine equal in eflficiency to the 
best reciprocating engines. 

Fig. 102 is a picture of one of theSe modern steam turbines. 
The cover has been removed so 
that we can see how it is made. 
Instead of a few large blades, 
like the water wheel, it has many 
rows of small blades fastened to 
a steel cylinder called a rotor. 
These movable blades pass be- 
tween rows of stationary blades 
fastened to the case of the ma- 
chine. High pressure steam en- 
ters the turbine at the smaller 
end of the case, and, in elbowing its way amongst the forest 
of blades, sets the rotor into rapid rotation. The arrangement 
of the fixed and moving blades in this turbine is shown in Fig. 103. 
The dotted line cemo indicates the path of the steam. 

Turbines have now been so far perfected that their efficiencies 
are greater than those of reciprocating engines. On account of 
their freedom from jarring, their high efficiency, and their com- 
pactness, they are now coming into general use. It is interesting 
to note that the principle of the steam turbine was known to Hero 
of Alexandria (b.c. 120). The technical difficulties involved 
in the practical construction of a steam turbine of high efficiency 
have delayed its perfection for 2000 years. 





Fig. 103. Blades in the Steam 


1. The mechanical equivalent of 1 gm cal is 4.19 X 10^ ergs. 

2. A gas is heated when it is compressed and cools when it 
does work in expanding. 

3. When a liquid evaporates, heat is absorbed. 

4. The work done by the steam in a steam engine is measured 


by the product of the average pressure and the volume of the 

5. The efficiency of an engine maybe increased (a) by raising 
the boiler temperature; (b) by using a condenser. 

6. The efficiency of a perfect engine is equal to the difference 
in temperature between the boiler and the condenser divided by 
the absolute temperature of the boiler. 


1. A tea-kettle of liquid air boils furiously when placed on a cake of 
ice. Does this case differ essentially from that of a kettle of water on 
a hot plate of iron? 

2. Describe the experiment of Joule and Rowland. What relation 
was established by them? 

3. With the aid of diagrams made from memory, explain the action 
of the steam in the cylinders of a steam engine, and the manner in 
which this action is controlled by the slide valve. 

4. Why is it possible to "ciit off" the entrance of the steam to the 
cylinder before the completion of the stroke, and still get work out of 
the steam that has entered? 

* 5. What is the use of the condenser of a steam engine? 

6. From the expression for efficiency in Art. 168, can you tell why 
the cylinder of a gas or gasoline engine is arranged so as to be cooled 
with water or air? 

7. Why should the rotor and case of the steam turbine, Fig. 102, be 
larger at the end where the steam leaves than it is at the end where it 

8. We can convert a given quantity of mechanical energy into 
heat. Can we convert a given quantity of heat entirely into mechanical 
work? Why? 


1. When a warm, moisture-laden wind strikes the sides of ^ moun- 
tain range, it is forced up the inclined plane, and rises to where the 
atmospheric pressure is less. What effect has this change of pressure 
on its volume? Since it is expanding against some atmospheric pres- 
sure, what effect does this expansion have on its temperature? What 
effect may this change of temperature have on the invisible water 
vapor that it contains? If the air then blows over the mountains, 
will it be likely to deposit much rain on the other side? 

2. How does Pascal's principle operate in the cylinder of an engine? 
Does equa.tion (9), Art. 122, apply there? 


3. Niagara Falls are about 53 m high. If all the energy of the fall- 
ing water were transformed into heat, how much would each gram 
heat itself by falling? 

4. The cylinders of a locomotive are 60 cm long and 50 cm in diam- 
eter. What is the volume of each? Take ir = 3.14. What volume 
of steam is used per stroke? The average effective pressure of the 
steam is found to be 3.5 X 10* dynes. How many ergs of work are 
done per stroke? If 3.3 strokes are made per sec, what is the power 
in -^^^? What is the horse-power? Remember that there are two 
cylinders, and that a complete stroke is twice the length of the cylinder. 

6. In June, 1892, in a test of the Empire State Express, the follow- 
ing data were recorded when the train was going at a speed of 60 ~-^. 
Find the horse-power developed. Area of each piston, 283.5 inches^; 
pressure, 53.7 ^^^^^^^r^J length of cylinder, 2 ft.; revolutions per 
minute of drivers, i.e., strokes of piston, 260. 

6. Fig. 104 is a pressure-volume graph. When the volume is in- 
creasing from pi to p2» does the pressure change? If the number of 
cm in the length of the ordinate Vj pi be multiplied by the number of 
cm in the length of V2 — Vi, what area does this product represent in 
the figure? Does this area also represent the work done by the 
gas in expanding from volume Vi to^ volume V2 ? Might we get the 
numerical value of this work by multiplying this area of the rectangle 
Vi pi P2 V2 by the number of dynes pressure and the number of cm'^ 
volume respectively, that 1 cm represents on the diagram? If the pres- 
sure changed to a different value pa, for a change of 
volume .V3— Vi, might we get in the same way 
the work done during this new period of expan- 

7. In the graph, Fig. 99, draw at equal dis- 
tances ten vertical lines from the atmospheric 
line p5 Pi to the broken line pi p2 Ps, measure 
these linesj and take their average. Will this 
average represent approximately the mean effec- 
tive pressure of the steam against the piston? . If 
we multiply this average by the length V2^^i» ^ ^z 
what area will it represent? Will this area rep- f 104 
resent approximately the work done by the 

piston in traveling from one end of the cylinder to the other? 

8. Fig. 105 represents the pressure gauge mentioned in Art. 163. 
The lower end of the tube at the left is coupled to one end of the engine 
cylinder, and opens into it; so the steam can enter and push up the 
little piston (seen inside the tube). This piston is held down by the 
spiral spring near the top of the tube, but when the piston is pushed 





up against this spring, it lifts the lever. The lever carries a pencil 
which it moves vertically along the little drum at the right of the dia- 
gram. A card is wrapped around 
this drum and held by the spring 
clips. A cord, which is wrapped 
around the drum, runs over a pulley 
at the right of the drum, and thence 
to a pin on the piston rod of the en- 
gine. As the piston moves back, the 
cord is unwound, and rotates the 
drum. When the piston moves for- 
ward, a spring turns the drum back 
again and winds up the cord. Can 
you apply your knowledge of the 
composition of motions so as to tell 
how the pencil draws on the card a 
pressure-volume graph like Fig. 99? 
This pressure gauge is known as the 
steam indicator. It was invented by 
James Watt. The graph is called by engineers an indicator diagram 
and is much used in determining the work done by a steam engine. 
The horse-power is obtained by dividing the work done per sec by 550 
{cf. Art. 43), and is called the indicated horse-power. 

9. What is the efficiency of an engine that consumes 1000 gm of 
coal per horse-power hour? 

10. An engine with an efficiency of 15 per cent does work at the rate 
of 746 X 10* ^^. How many gm of coal must it burn per hour 
{cf. Art. 167)?^^ 

11. In a water turbine, moving water turns the wheel and the 
wheel turns machinery. Mention a case where a similar wheel is turned 
by an engine and made to produce motion by pushing against the 
water. A steam turbine and a windmill are used to propel machinery. 
Mention a familiar case of a similar wheel that is turned by a motor 
or engine and made to put air into motion. 

Fig. 105 


1. Hammer a nail rapidly for several minutes on a piece of iron, 
pick it up and see what happens. 

2. If you have a toy engine, bring it in and explain how it works. 

3. Visit a roundhouse or engine shop and make a report of what 
you see that interests you. Visit an ice and cold storage plant and 

4. Find out what you can about Count Rumford and his studies 
of the conversion of work into heat. 


5. You will find interesting information about heat in the follow- 
ing books: Mach, Heat, Open Court Publishing Company, Chicago; 
D. E. Jones, Heat, Light and Sound, Macmillan, New York; The Twen- 
tieth Century Locomotive, Sinclair Company, New York. A good 
description of the indicator diagram will be foimd . in PuUen's Me- 
chanics^ pp. 253-260. You will find much interesting inforrfiation 
concerning new forms of engines in the "Scientific American" and its 
Supplement, in "The Engineering News," "The Technical World," Cas- 
sell's and Cassier's magazines, and in "Railway and Locomotive En- 

6. Read the Life of James Watt, by Andrew Carnegie; the Life of 
Robert Fulton, by R. H. Thurston. An interesting account of the de- 
velopment of the steam engine is given in A History of the Growth of 
the Steam Engine, by R. H. Thurston. Who was Count Rumford? 
Read Memoir of Sir Benjamin Thompson, Count Rumford, by G. E. 
Ellis (American Academy of Arts and Sciences, Boston). 

7. The following books may interest you: Balfour Stewart, The Con- 
servation of Energy (Appleton, N. Y.); Ray S. Baker, The Boys* Book 
of Inventions, and the Second Boys* Book of Inventions (Doubleday- 
Page, N. Y.); R. H. Thurston, Heat as a Form of Energy (Houghton- 
Mifflin, Boston); T. O'Connor Sloane, Liquid Air and the Liquefaction 
o/(?ases(N.W. Henley, N. Y.). . 

8. A great deal.of useful information about all kinds of engines and 
fuels is given in Wm. Kent, Mechanical Engineer's Pocket Book (Wiley, 
N. Y., 6th Edition, 1903). 


172. The Transmission of Power. In the preceding chap- 
ters, we have seen how the energy of wind, of water, and of steam 
may be utilized in windmills, water-wheels, and steam engines 
for doing mechanical work. When the energy of these contrivances 
is transmitted by means of belts, cables, gear wheels, or shafting 
to the machinery which it is to operate, this machinery and the 
source of its energy must be near together. The eflBciency of 
such a system of transmission is not so great as could be desired, 
and it diminishes rapidly as the distance between is increased. 
Furthermore, the practical difficulties in the way of these methods 
of transmitting power become prohibitive at comparatively short 
distances. When the work to be done is that of transportation, 
the engine is a locomotive, and in doing its work it goes over any 
distance desired; but it must carry with it a heavy load of fuel and 
water, and it is not nearly so efficient for light loads and frequent 
stops, as for heavy loads and without stops. Is there no form of 
energy that may be transmitted cheaply and conveniently over 
considerable distances, and used at such times, at such places, 
and in such amounts as may suit the convenience of the user? 

Most readers know in a general way that there are two meth- 
ods by which energy may be thus distributed. One is by con- 
verting the coal into fuel-gas, and sending it through pipes to the 
various places where it is to be used for light, for heat, and for 
operating gas engines for power. The other is by converting the 
energy of the water-wheel or the steam eri^ne into that of an 
electric current, and distributing it by means of copper wires to 
electric lamps for light, and to electric motors for power. 

173. Electric Generators. Plate VII is a photograph of one of 
the large dynamo-electric machines of a power plant, whence 
electrical energy 13 distributed. It is direct-connected with a 






big compound steam engine. The engine transforms heat energy 
into mechanical energy; and the dynamo transforms the mechan- 
ical energy into electrical energy. This electrical energy is sent 
out along a system of wires to the places where it is to be used. 
These powerful machines suggest many interesting questions 
for study, some of which we shall try to answer in the next four 
chapters. How do these machines work? What are some of the 
elementary facts of electricity and of magnetism? What are the 
relations between electricity and magnets? What are some of 
the useful inventions by means of which the discoveries in elec- 
trical science are applied so as to multiply both our means of 
doing business and our facilities for enjoying Hfe? Who were 
some of the great discoverers that sowed the seeds from which 
this harvest has sprung? 

174. Early Knowledge of Electricity and Magnetism. Be- 
fore history began, man feared the lightning and thunder, and 
had some primitive explanation for it. Amber, when rubbed 
with wool, attracts light bodies. Doubtless the fact was known 
long before Thales of Miletus recorded it, about 600 B.C. The 
early Greeks also knew that lodestone, or magnetic ore of iron, 
attracts pieces of iron; and some truth and many extravagant 
fables were written of it by Pliny and others. The magnet is 
supposed to have received its name from Magnesia, in Asia Minor, 
where deposits of lodestone were found. The word "elec- 
tricity" is derived from "electron," the Greek name for amber. 

The Greeks never used the scientific method of study, and 
hence they learned nothing of electricity beyond a few simple 
facts, which in themselves are useless. They contented them- 
selves with supposing that amber possessed a soul, which gave 
it its strange powers. 

176. Oilbert. The first man who ever studied electricity and 
magnetism to any purpose was William Gilbert of Colchester 
(1540-1603), a contemporary of Shakespeare and Bacon. Queen 
Elizabeth appointed him her physician, and gave him a salary, 
in order that he might be free to pursue his studies. He collected 
and recorded all that was then known about the subject; and as 


a result of his experimental studies he discovered many new 
facts. These he published in 1600 in a book, De Magnete, which 
is still of good scientific repute. 

176. Electrification. Gilbert found that other substances 
besides amber, such as glass, sulphur, and the resins, would, 
when rubbed, attract light bodies. When in this condition, they 
are said to be electrified or charged with electricity. He 
found also that he could not charge metals by rubbing them; 
therefore he called the former electrics and the latter 7ion' 

177. Condnctors and Insnlators. Gilbert's lack of success 
with his "non -electrics'' was due to a very important electrical 
property, of which he failed to leam, but which was discovered a 
century later by another Englishman, Stephen Gray. This 
property, now so well known, is called conduction. An electric 
charge will travel over or through some bodies very easily, some- 
what as heat does; and it can thus be transmitted along them 
from one place to another. Accordingly, these substances are 
called GOOD conductors of electricity. Those substances which 
do not conduct well are called poor conductors or insulators. 
As a result of experiments, substances may be arranged as below, 
in the order of their eleqtrical conductivities. 

CONDUCTORS It is worthy of note that for most 
Silver substances the order of their electrical 
Copper conductivities is the same as that of 
Iron their heat conductivities. There is no 
Mercury dividing line between conductors and in- 
Carbon sulators; for every substance has some 
Solutions of Salts conducting power. The difference is 
Pure Water merely one of degree. 
Resins We ought now to know how to 
Hard Rubber succeed where Gilbert failed. To elec- 
Porcelain trify a piece of metal, we have only 
Glass to fasten it to a handle made of one of 
INSULATORS the substances near the foot of the list; 
and we shall find that we can charge it as highly as we can any 


of the others. The human body and the earth are fairly good 
conductors; and if it were not for the insulating handle of glass 
or rubber, the charge would escape through the body of the ex- 
perimenter, and spread itself over the earth. Consequently, 
there would be so little energy left at any one place that no per- 
ceptible work would be done by it. 

178. Bepnlsion. That an electric charge may cause repul- 
sion as well as attraction was first noticed by Guericke, who 
devised the first electrical machine as well as the first air pump. 
When two pith balls suspended by threads from an insulating 
support are approached by an electrified glass rod, the rod at- 
tracts the balls to itself. If the balls touch the rod, some of the 
charge from the rod is communicated to them, and they are re- 
pelled. The balls now repel each other; they also repel the rod, 
as we should expect from the third law of motion (c/. Art. 40). 
This may easily be shown by suspending the rod at its middle by 
a paper sling, attached to a silk thread. In the repellent move- 
ment, the rod, of course, has a smaller acceleration than have 
the balls, because its mass is greater. 

179. Discharge. If now the pith balls are touched by the hand, 
or by any other conductor connecting them with the earth, their 
charges will be dissipated. When this has occurred, they are 
said to be discharged. 

180. Two Kinds of Electrification. If the balls are both 
charged from any other electrified body, they will repel each 
other just as they did When charged from the glass. We find, 
however, that their electrification is not always of the same sort; 
for if we electrify them by contact with glass that has been rubbed 
with silk, and then present to them a stick of sealing wax that 
has been rubbed with flannel, we observe that though there is 
repulsion between the balls and the glass, there is attraction 
between the balls and the wax. Thoroughly discharge two pairs 
of pith balls, A and B, electrify the pair A from the glass, and 
the pair B from the wax. Those of the pair A repel each other; 


those of the pair B repel each other. But those of the pair A 
attract those of the pair B, and are attracted by them. It was 
thus found that there are two kinds • of electric charges, and it 
became necessary to name them. That kind of charge which is 
developed on glass by rubbing it with silk, is called a vitreous 
or + (positive) charge; and that kind which is developed on 
sealing wax by rubbing it with flannel, is called a resinous or — 
(negative) charge. These names are, of course, purely arbi- 
trary, and are adopted solely for convenience. 

181. Bnfay's First Law. The results of experiments like 
those just described are included in the following general state- 
ment, which we may call the law of electrostatic attractions and 
repulsions, or the first law of electrostatics. It is also known by 
the name of its discoverer, Dufay. 

Like charges repel each other; unlike charges attract each other. 

182. To Determine the Kind of Charge. Since a charged 
body always attracts an unelectrified body, as well as one having 
a charge of opposite sign, attraction is not a satisfactory test of the 
kind of charge that a body has; but if repulsion occurs between 
two bodies we may be sure that they have like charges. If, then, 
we wish to determine the sign of an unknown charge, * we may 
impart some of this charge to a pith ball, and place the ball first 
near a glass rod rubbed with silk, then near a stick of sealing wax 
rubbed with flannel. If the ball is repelled by the glass rod, the-- 
unknown charge is of the positive kind; and if it is repelled by the 
wax, the unknown charge is of the negative kind. 

183. Electroscope. A suspended and insulated pith ball 
thus serves as an electroscope, by means of which we may detect 
a charge, and determine its sign; but for many experiments a 
more sensitive instrument is needed. The one shown in Fig. 106 
answers well. Two strips of gold or aluminum leaf take the 
place of the pith balls; and the flask serves for an insulating sup- 
port, as well as to protect the light and fragile leaves from any 
disturbing currents of air. A very slight charge communicated 



to the metallic ball or plate at the top is conducted to the leaves, 
and causes them to repel each other. Also, the greater the charge, 
the greater the divergence of the leaves. When the ball or plate 
is touched by the hand, the leaves collapse, showing that the 
electroscope is discharged. 

To test a charge by means of the gold leaf electroscope, give 
the leaves a known charge suflBcient to cause a moderate diver- 
gence. If now a charge of 
the same sign is approached, 
the divergence of the leaves is 
seen to increase ; but if a charge 
of the opposite sign is ap- 
proached, their divergence is 
seen to diminish.' Therefore, if 
we have given the electroscope, 
say, a negative charge, and if 
we then bring near it the un- 
known charge, an observed in- 
crease in .the divergence of the 
leaves will prove the unknown 
charge to be negative, and a decrease in their divergence will 
prove this charge to be positive. The proof plane P, Fig. 
106, is a disc of metal with an insulating handle. It is used to 
carry a small charge from a charged body to the electroscope 
in order to test the body's electrical condition. 

184. Both Snbstances Charged. We may now ask, Is it 
likely that the substance with which we rub the glass or the wax 
suffers no change of condition? Ought we not to suspect that it 
also receives a charge? And will the charge, if it has one, be of 
the same kind as that of the substance rubbed, or of the opposite 
kind? For answer, rub the glass and silk together. When 
tested with the electroscope, the silk will prove to be negatively 
electrified. Rub the wax and flannel together; and the flannel, 
when tested, will prove to be positively electrified. In this ex- 
periment, the silk and the flannel must be tied to insulating han- 
dles of glass or rubber, as they themselves are not sufficiently. 

FiQ. 106. The Electroscope 



good insulators to retain their charges when held in the hand. 
In this way it has been shown that whenever two dissimilar sub- 
stances are rubbed together, one gets a positive charge and the 
other a negative charge. 

186. The Two Charges are Eqnal. But what about the rela- 
tive amounts of the electric charges of the glass and the silk, or 
the flannel and the wool? Are they equal or unequal? We may 

call them equal if they pro- 
duce equal effects, or if one 
exactly neutralizes the effect 
of the other. Let us make an 
experiment which, though 
rather crude, is nevertheless 
convincing. Two brass discs, 
A and 2?, Fig. 107, are fas- 
tened to insulating handles. 
The one on the right, in the 
picture, is faced with a disc of 
flannel or fur, of exactly its 
own size, and neatly ce- 
mented on with sealing wax. 
First thoroughly discharge both discs and the electroscope. 
Fit the discs accurately together, face to face, and twist one of them 
half way around and back again, so as to rub them together. 
Hold A two or three centimeters from the electroscope, and note 
the amount of divergence of the leaves. Withdraw A, and put 
B as nearly as possible in the same place. The leaves are seen 
to diverge; and the divergence is the same in amount as before. 
Now, without having allowed the discs A and B to touch 
anything, fit them accurately together again; and while they 
are thus held, bring them near the electroscope. Observe 
that while they are together there is no effect on the leaves 
of the electroscope, i.e., the two opposite charges exactly neu- 
tralize each other's effects. Thus we know that they are equal 
in amount. 

These matters have been thoroughly and accurately tested 

Fig. 107. The Charges are Equal 


by many experiments, all of which go to prove the following 
general statement: 

Any two dissimilar substances when brought into intimate con- 
tact and then separated, acquire equal electrostatic charges of opposite 

186. Pranklin's Theory. One of the most noted discoverers 
in the field of electrostatics in the eighteenth century was. our 
own Benjamin Franklin (1706-1790). Franklin proposed a 
theory, which in his own time was very widely accepted. He 
supposed that all unexcited bodies have an indefinite supply of 
electricity, which is of one kind only, and that charges are gen- 
erated by one body getting some of this electricity from some 
other body, so that the former has an excess of electricity, while 
the latter has an equal deficiency. The body having the excess 
was said to have a positive charge, and the other an equal negative 
charge. If was Franklin who proposed the use of the positive 
and negative signs as suggested by this theory. 

This theory is very useful as a working hypothesis. With 
some slight modifications as to ideas and terms, it is still com- 
petent to describe all electrostatic phenomena, including some 
very remarkable ones recently discovered. 

187. Electrostatic Polarization. Let us now see how we 
can use this theory as a working hypothesis for the description 
of phenomena and the discovery of new facts. In experimenting 
with the electroscope, we can hardly have failed to notice a re- 
markable fact which requires explanation, namely, that the leaves 
diverge widely whenever an electrified body approaches the instru- 
ment. This divergence occurs although the electrified body does 
not touch the electroscope, nor even approach it near enough 
for a spark to pass. Using our hypothesis, we may say, in explan- 
ation, that the + electrification of the glass rod, when brought near 
the uncharged electroscope, causes a disturbance of the neutral 
electrification of the electroscope; it repels positive (+) elec- 
trification into the end farthest away, and an equal negative 
(— ) charge remains at the nearer end. The leaves, there- 



fore, being both positively charged, repel each other. In this 
way, any neutral body may be given a + charge at one end, and 
a — charge at the other, both of these charges being equal in amount 
to the charge that causes this redistribution. A body in this 
condition is said to be electrostatically polarized. Fig. 108, a. 
That a conductor is really in the condition just described, when 
under the influence of a near-by charge, may easily be shown by 
taking small portions of its charges on a very small proof plane (P, 
Fig. 106), and testing them with the electroscope. We can thus 
prove that the end nearest the influencing charge has a charge of 
opposite sign, that the other end has a charge of the same sign as 
that of the influencing body, and that the middle region is neutral. 

188. Orounding the Repelled Charge. If the influencing 
charge be removed from the neighborhood without our having 
touched the electroscope, the latter returns to the neutral 
condition, as is evidenced by the collapsing of the leaves. We 
may say, then, that the opposite charges at its two ends have 
united and neutralized each other. But if we touch the plate 

+ +■».+ + 4. 4. i + + •!• + 


X \ 


I \ 

* A- 

b "'^^'^E c '^ d 

Fio. 108. Charging by Induction 
a. Polarized, b. Grounded, c. Bound charge, d. Charged. 

with the hand while the influencing charge is held near, the theory 
leads us to expect that the repelled + charge, in trying to get as 
far as possible from the glass rod, will go to the earth through the 
body, while the — charge will remain on the plate, bound by 
its attraction for the + charge on the glass rod. That the re- 
pelled charge does go from the leaves to the earth is indicated by 


the fact that as soon as we touch the electroscope plate so as to 
connect it with the earth E, Fig. 108, the leaves collapse. Allow- 
ing a charge thus to pass to the earth along a .conductor is often 

called GROUNDING THE CHARGE, Fig. 108, b. 

189. Charging by Influence. The theory now suggests an- 
other step. When we have polarized the electroscope and grounded 
the repelled + charge, suppose that we remove the earth connec- 
tion before we remove the influencing charge. Will not the 
electroscope then be left with a — charge? It would seem so, be- 
cause breaking the earth connection would certainly prevent the 
return of the + charge, while the — charge would be held by its 
attraction for the influencing charge. Fig. 108, c. When this bound 
charge is released by removing the influencing charge, it will 
move freely over the electroscope and reveal its presence by 
causing a divergence of the leaves, Fig. 108, d. That this excess of 
electrification is really present and is of the negative sort, may 
easily be shown by the fact that the leaves collapse when the posi- 
tively charged glass rod is brought near, and increase their 
divergence when approached by a negative charge. This is the 
most convenient method of charging the electroscope. Any con- 
ductor whatever may be charged in this way. This kind of 
charge is called an induced charge, and the process by which 
a conductor B is electrified by another body A without loss 
by A of any part of its own charge, is called charging by 
influence. Let us review the steps of the process. They are: 

1. Bring the influencing charge A near' the neutral conductor 
B; B becomes polarized. 2. Touch B with the hand, or with 
any earth-connected conductor; the repelled charge, equal in 
quantity to A, and the same in kind, is grounded. 3. Break the 
earth connection; the grounded charge can not get back, and the 
bound charge, equal to A and opposite in kind, remains. 4. Re- 
move A; the bound induced charge is freed, and spreads over 
the surface of B, That the induced charge is equal to the 
influencing charge may be proved, if both bodies, A and B, are 
good conductors, by allowing A to touch B, when the two charges 
will unite and neutralize each other. That the two charges are 


opposite in kind may be proved by testing them with the electro- 

190. A Static Charge Besides on the Outside of the Con- 
dnctor. When a charge is communicated to a conductor, any 
two portions of it will repel each other; and therefore every por- 
tion of the charge will get as far away from every other portion 
as it can. Thus we should expect to find a charge distributed 
over the surface of the conductor, and not at all on the inside. 
That this is true, may be proved by using a very small proof 
plane (Art. 183), with which to take samples of the electrification 
from the different parts of the conductor. Thus we may electrify 
an insulated, hollow brass globe, or even a common tin cup placed 
on a cake of resin. With the proof plane, we may then test all 
places on the outside of the hollow conductor, carrying to the 
electroscope the sample charges, which will reveal their presence by 
the divergence of the leaves. But try as we may, provided the 
proof plane is not allowed to come too near the edges of the open- 
ing in the hollow conductor, we can not succeed in getting any 
charge from the inside. 

191. Conlomb's Law. The magnitude of the force between 
two charged bodies was first determined in 1777 by Coulomb, 
an eminent French engineer arid physicist. He measured this 
force by balancing it against the torsion (twisting force) of a 
fine wire. He found that when the distance between two charges 
was made twice as great, the force was reduced to J its former 
value; and when the distance was made four times as great, the 
force was reduced to yV- Coulomb's experiments also proved 
that if either charge was increased in amount, the force in- 
creased in the same proportion. The facts established by such 
experiments may be summarized in the following general state- 
ment, which we may call Coulomb's law of electrostatic force, 
or the third law of electrostatics. 

The force between two electrostatic charges varies directly cw 
the product of their quantities, and inversely as the square of the 
distance between their centers. This law is strictly true only when 


the charges are situated on spherical conductors, which are very 
small in proportion to the distance between them. This state- 
ment furnishes us with an appropriate unit in which to measure 
a charge, for we may define unit charge as thai charge which, 
placed in air at a distance of 1 cm from an equal charge of like 
sign, repels it with a force of 1 dyne. 

192. The Leyden Experiment. In the year 1745 a discovery 
was announced from Germany, and a few months later from 
Leyden, in Holland, which caused experimenters to redouble 
their activity. While experiments were being made in electrify- 
ing water in a bottle held in one of his hands, Musschenbroek, 
a renowned science teacher of Leyden, touched the wire by 
which the charge was passing into the water, with his other hand. 
He receiyed a muscular shock of extraordinary power. The news 
of the experiment spread rapidly, astonishing all Europe, and it 
was repeated everywhere, both in Europe and in America, with 
dramatic effect. At the French court, birds and small animals 
were killed by electricity; and 180 soldiers in line were given a shock 
simultaneously. A line of Carthusian monks 900 feet long was 
formed, and the dignified ecclesiastics were made to jump up all 
together, by the discharge of the ''Leyden bottle." All this 
partook rather more of the spectacular than of the scientific; 
but it served a good purpose, first in arousing general interest 
in scientific experiments, and second in providing physicists 
with a new combination to investigate, and a means of collecting 
greater charges than had previously been at their disposal. 

193. Condensers. The essential parts of the Leyden apparatus 
are: 1. Two conductors of large surface; 2, a thin layer of 
some good insulating substance between the conductors. The 
two conductors are called the coatings, and the insulating layer 
is called a dielectric. Such an arrangement of two conductors 
and a dielectric is called a condenser. It soon took the familiar 
form of a glass jar, coated inside and out with tin foil, reaching 
to within 5 or 10 cm of the mouth. The mouth is closed with an 
insulating stopper, through which passes a brass rod, terminated 



above by a brass ball, and below by a chain which touches the 
inner coating. This is called a leyden jar. » 

194. To Oive a Condenser a Large Charge, one coating must 
be connected with one of the knobs of an electric machine, and 
the other coating with the other knob or with the earth. As the 
machine is operated, and the electrical energy given out, this energy 
is stored in the condenser. To discharge the condenser , the two 
coatings must be joined by a conductor. When the conductor 
has almost completed the circuit, the air between the knob of 
the condenser and the end of the discharger breaks down, or is 
punctured, and a spark passes, accompanied by a loud snapping 
noise. This is known as a disruptive discharge. Such sparks 
are shorter, thicker, hotter, noisier, and in every way more ener- 
getic than those which the machine can give without the con- 

195. How the Condenser Operates. Since each of the coatings 
of the condenser is connected with one of the poles of the electric 
machine, one coating becojnes charged 
positively and the other negatively. 
These two charges are not able to 
neutralize each other's effects, because 
of the dielectric between them. They, 
however, strongly attract each other, 
and hold each other bound on the oppo- 
site surfaces of the dielectric. Thus 
the charges cling to the two sides of the 
dielectric. This fact was discovered by 
Franklin, and may be easily proved by 
means of a Leyden jar, whose inner 
and outer coatings can be removed. 
Such a jar is shown in Fig. 109. After 
the jar has been charged, the two coatings are removed and 
discharged. When they are replaced, the jar will be found to have 
retained its charge, for a bright spark may be obtained from it. 
Hence the charge remained bound on the two sides of the 

Fig. 109. The Charge is in 
THE Dielectric 


dielectric, even after the outer conducting coatings had been 

196. The Dielectric is in a Strained Condition. It has been 
found useful to conceive that when a condenser is charged, the 
dielectric between the two charges is in a strained condition. 
This idea was first suggested by Faraday. The discharge of 
the jar, then, consists merely in the release of the strain. Fara- 
day also enlarged our conceptions of condensers by showing that 
whenever a conductor is charged, an equal, opposite charge is 
induced on some neighboring conductor, or on the walls of the 
room. The dielectric between these two charges is in a state of 
strain. Hence every charge, from that of a little pith ball to that of 
a thunder cloud, may be regarded as the charge of a condenser; 
for there are always two equally and oppositely charged conductors 
separated by a dielectric. 

197. A Disruptive Discharge is Oscillatory. Since we have 
seen that an electric charge always implies electric strains in the 
medium between the two oppositely charged bodies, we might 
suspect that a discharge consists in the release of this strain. 
We may get a rough mechanical picture of the state of things by 
considering two plums, imbedded in a mass of elastic gelatin. 
If the two plums are separated by stretching the gelatin, they tend 
to come together again and resume the positions which they 
had before the gelatin between them was strained. If we release 
them by allowing them to slip back gradually, they will cease to 
move when they reach the position of no strain. If, instead of 
releasing them gradually, we release them suddenly, they first fly 
beyond their positions of equilibrium, and then fly back nearly 
to their starting points. So they continue to swing, or oscillate, 
back and forth, but through smaller and smaller distances, until 
they come to rest finally in their normal positions. Now, if an 
electric discharge is the releasing of a strain in an elastic medium, 
as we have conjectured, we can see from the consideration of the 
crude gelatin model that when the spark passes, an oscillation 
of some sort must take place. These electric displacements. 



first in one direction and then in the opposite direction, may be 
conceived to be electric charges of opposite sign; and a disruptive 
discharge would then consist of a rapid surging movement, or 
alternating current, between the two conductors. If Faraday's 
elastic displacement theory is anywhere near the truth, and if we 
have reasoned correctly from it, we ought to conclude that the 
spark from an electrostatic machine, or from a condenser, is 
oscillatory, and it ought to.be possible to prove it by actual ex- 
periment. It has been shown mathematically that the oscillations 
which make up the discharge «,re exceedingly rapid. Neverthe- 

FiQ. 110. The Spark Oscillates 

less, it has been possible to photograph them and determine their 
period, by means of a rapidly vibrating or rotating mirror. The 
periods of • oscillation of the sparks from Ley den jars depend 
mainly on the sizes of the jars, and range from one thousandth 
to one ten-millionth of a second. Fig. 110 is a photograph of 
such an oscillatory discharge. The mirror was turned to the left 
while the charge surged alternately up and down between two 
brass balls. The mirror, as it passed, threw an image of each 
surging on to a photographic plate and since the mirror turned a 
little each time, the successive images were thrown on different 
parts of the plate. 

198. Lightning. During his extended experiments with con- 
densers and their powerful effects, Franklin began to be impressed 
with the many resemblances between condenser discharges and 
lightning. Conjectures of this kind had been advanced from 
time to time by European physicists, from the days of Gray, but 
in the mind of Franklin it had grown into a conviction. In 1749 
he stated his conviction, gave his reasons for it, and proposed an 



Fio. 111. The Electric Spark in the Labo- 

experiment with a pointed aerial wire by means of which the 
electricity might be conducted quietly from the clouds to the 
earth. His proposition 
was received with skep- 
ticism or indifference, 
except at the French 
court, where one exper- 
imenter tried it and 
was successful. 

Franklin, hotvever, 
did not regard this experiment as conclusive, because the 
wire did not reach into the clouds; and he therefore thought 
that the charge might have been received in some other way. 
Accordingly, he devised an experiment which, for boldness of 
conception and dramatic interest, as well as for its conclusive 
character, stands unsurpassed in the history of electrical 
research. He made a kite from a large silk handkerchief j and 
tipped it with a pointed wire. He awaited a thunder storm, 
and went out with his son to fly the kite. After sending the 
kite up into the overhanging rain cloud, he held it by a strip 

of silk attached to the 
hemp twine, and waited 
calmly for the appear- 
ance of the sparks, 
which he hoped to re- 
ceive from a metal door- 
key attached to the 
string. " No man,' ' 
says a modem writer, 
"ever so calmly, so 
philosophically, staked 
his life upon his faith.'* 
He believed that the 
pointed wire would 
bring, not a disruptive 
discharge, which he knew would kill him if it came, but a quiet 
flow down the kite-string. At first there was no result; but after 

The Electric Spark in Nature 


it had begun to rain, and the string had become wet, the hempen 
fibers began to bristle up, and sparks came in plenty from the 
key. A Leyden jar was charged, and the charge was thoroughly 
tested for all the properties of electricity. Thus were the light- 
nings and the thunders of Nature's great laboratory identified 
with the sparks and the snappings of the machine in the philoso- 
pher's laboratory. Thus were the valuable researches of **the 
many-sided Franklin" crowned by a great discovery. Its pub- 
lication produced a profound sensation in Europe as well as in 
America. The honors which his researches brought him were 
well deserved, for he was the greatest experimental philosopher 
of his time. 

In closing this chapter, it may be remarked that the benefit 
TO MANKIND resulting from the study of electrostatics, up to the 
point that we have now reached, was almost wholly intellectual. 
Electrostatic phenomena are almost entirely devoid of practical 
applications; but the intellectual progress which resulted from 
the experimental study of these phenomena prepared the way 
for discoveries of the phenomena of electricity in motion, 
and for the vast throng of important inventions which depend 
upon them. Some foolishly "practical'' person once asked Dr. 
Franklin what use might be made of the facts proved by some 
of his experiments. The great philosopher replied pithily by 
asking him. What is the use of a baby? The discovery of the 
oscillatory character of the condenser discharge illustrates the 
force of Franklin's answer to this much asked question. What 
is the use of a scientific discovery? In the course of time, 
Maxwell deduced from theory that the ether (cf. Art. 148) might 
be the medium in which electrostatic strains and oscillations take 
place; and that if so, the discharge of a condenser must 
start waves in the ether. Hertz went to work to start 
such waves, and detect them. He succeeded. Then followed 
the discoveries of more sensitive detectors by Branley and Lodge, 
and the great work of the inventors in wireless telegraphy, 
which is now going on. Like a baby, a scientific discovery 
may be small and uninteresting to many; but no one can tell 
how important it may become. 



1. Mechanical energy may be converted into electrical energy. 
This electrical energy may be economically transferred to distant 
places, and there reconverted into mechanical energy, heat or 

2. In order to understand how this is done, one must know 
the elementary facts and laws of electrical phenomena. 

3. Amber and sealing wax, when rubbed with wool or fur, 
attract light bodies/ and exhibit other remarkable properties. 
They are then said to be charged with electricity. 

4. Electrification travels along some substances easily, but 
along others with great difficulty. The former are called con- 
ductors, the latter, insulators. 

5. Both conductors and insulators are essential to the trans- 
ference of electrical energy. 

6. There are two kinds of electrification, positive and nega- 

7. Like charges of electricity repel each other; unlike, attract. 

8. Any two dissimilar substances, when brought together and 
then separated, become equally and oppositely charged. 

9. The theory of Franklin affords a convenient language for 
the description of electrostatic phenomena. 

10. A neutral body may be. electrostatically polarized, and 
charged by influence. The induced charge is equal to the in- 
ducing charge, and opposite in kind. 

11. A static charge does not exist anywhere inside a closed 
conductor; it is always found on the outside. 

12. The electrostatic unit quantity of electricity is that quan- 
tity which, placed in air at a distance of 1 cm from an equal quan- 
tity of like sign, repels it with a force of 1 dyne. 

13. The repulsive or attractive force between two charges 
is directly proportional to the product of their quantities, and 
inversely proportional to the square of the distance between them. 

14. A condenser consists of two conducting plates with a thin 
layer of dielectric between. 

15. The charge of a conductor is in the dielectric, not in the 


16. Franklin proved that lightning is a disruptive electrical 
discharge from cloud to cloud, or from cloud to earth. 

17. The dielectric between two charged conductors is in a con- 
dition of strain. 

18. Every disruptive discharge is a condenser discharge. 

19. A disruptive discharge is oscillatory and starts ether waves. 


1. What great advantage has electrical energy over other forms? 

2. What is the most obvious property of a body that is charged 
with electricity? 

3. What are conductors and insulators? Why are both necessary 
to electrical transmission? 

4. Why do we say that there are two kinds of electrification? How 
are the two kinds named? How may the presence and the kind of 
charge be determined with the aid of the electroscope? 

5. With the aid of Franklin's theory, explain how a conductor may 
be polarized, and charged by influence. How do the induced, and 
inducing charges compare with each other in kind and amount? 

6. Define the electrostatic unit of quantity. 

7. What are the essential parts of a condenser? How is it charged 
and discharged? Describe the manner in which the charge is accumu- 
lated, accounting for the large capacity, i.e., ability to hold a large 

8. What is a dielectric? Does the charge of a condenser belong 
to it or to the coatings? How may this fact be shown? 

9. What is the physical condition of the dielectric between two 
charged conductors? 

10. Describe a disruptive discharge. Explain why every charged 
body must be regarded as one. of the coatings of a condenser. 

11. What is the peculiarity of a disruptive discharge? 

12. How may the heat, light, and noise of a disruptive discharge be 
accounted for? 


1. Two unlike electrostatic charges of Q and Q' units, respectively, 
are distant d cm from each other. Will they attract or repel each 
other? Call their force in dynes /, and write the expression for its 

2. A certain charge is placed at a distance of 10 cm from a-f- charge 
of 250 units, and the two charges are found to repel each other with a 
force of 10 dynes. What was the amount and sign of the unknown 


3. In Fig. 108, a, suppose the charge of the glass rod is 10 units, and 
its distance from the plate of the electroscope is 2 cm. How many — units 
are attracted to the plate? How many -|- units are repelled to the 
leaves? Suppose this + charge to be 10 cm from the repelling charge. 
With how many dynes force is it repelled? What is the amount and 
direction of the resultant force between the rod and the electroscope? 
Do your answers suggest a reason why a charged body attracts an 
uncharged body? 

4. Does the self-repulsive property of a charge suggest why it is 
found that when a body with sharp points is electrified, a large pro- 
portion of its charge is collected at the points? It is also found in 
such cases that the charge escapes rapidly from the highly charged 
points, with streams of air that flow away from each of the points, 
sometimes with force enough to blow out a candle. From your knowl- 
edge of electric attraction and repulsion and of electrifying by contact, 
explain how such streams of air are maintained until the body is dis- 

5. Does question 4 suggest the reason why bodies intended to hold 
electrostatic charges are usually made round and smooth? Does it 
suggest why an electrostatic machine usually refusfes to "spark" when 
a pointed wire is attached to it, or grounded and brought near it? 
Does it show grounds for Franklin's belief that the charge from the 
pointed wire on his kite would not harm him? 

6. It is found that the electrostatic capacity of a condenser, i.e., 
its ability to accumulate a large charge under given conditions, is di- 
rectly proportional to the area of its coatings, inversely proportional 
to the thickness of the dielectric between the coatings, and also depends 
on the material of the dielectric. If a charge is of the nature of a 
strain in the dielectric, can you see a reason for each of these three 

7. The dielectric of ^ condenser is often found to have a residual 
charge, i.e., a second discharge is obtained from it when a little time 
has elapsed after the first. Does this fact indicate that the charged 
dielectric is in some such condition electrically as an elastic body is 
mechanically when it is compressed, stretched, or twisted? 

8. Can you show that an electrostatic charge has potential energy, 
like a strained spring? What work is done to store this energy? 


1. Read Tyndall's Elementary Lessons in Electricity (Appletons, 
New York) for a fascinating account of electrostatic phenomena, with 
many practical hints for those who wish to make experiments inex- 
pensively for themselves. Read also Hopkins's Experimental Science^ 
pp. 359-391. 


2. Find out what you can about the early experimenters in the 
field of electricity, especially Gilbert, Franklin, and Faraday. The 
following books contain much that may interest you, and that you 
can find easily if you will consult their indexes : Benjamin's The Intel- 
lectvxil Rise of Electricity (Longmans, New York); Benjamin's The Age 
of Electricity (Scribners, New York) ; Cajori's History of Physics (Mac- 
millan, New York); Arabella B. Buckley's A Short History of Science, 

If you have access to a good cyclopedia, consult it often for further 
information about the men and the subjects mentioned in these pages. 
The Encyclopedia Britannica is especially strong on the side of science. 

3. If you will repeat for yourself the experiments mentioned in 
this chapter, and as many of those made by your teacher as you can, 
you will find that by thus becoming somewhat of an independent ex- 
perimenter, you will not only increase your ability to understand the 
lessons of this course, but you will also get a great amount of pleasure 
out of it, and acquire a kind of skill that may be of great service else- 



199. Lodestone and the Compass. Having learned in the 
last chapter some of the fundamental facts concerning electric 
charges and their properties, we will now take the next step 
toward finding out how the dynamo operates; and seek to dis- 
cover what a magnet is, what an electric current is, and what 
relations exist between electricity and magnetism. 

It is interesting to note in the first place that the attraction 
of a magnet for iron has been known from time immemorial; 
for magnetic iron exists in nature in the form of the mineral called 
magnetite, or lodestone. This mineral was known to the Egyp- 
tians and Greeks long before the Christian era, for in their writ- 
ings we find them speculating about its attraction for iron. 

Besides the attraction of lodestone for iron, little was known 
of magnetism, and no use was made of it, until the introduction 
of the compass in Europe by the Arabs, about the year 1200 
A.D. A COMPASS, as is well known, consists of a small strip of 
steel, which is first magnetized by rubbing it from end to end 
on a lodestone, or any strong magnet, and is then suspended on 
a pivot so that it is free to turn in a horizontal plane. Such 
a magnetic needle always tends to set its length in a north 
and south direction, and therefore it has been used by all 
civilized people for determining the north and south line. Since 
every magnet, when freely suspended, like the compass needle, 
settles in a definite position, which is nearly north-south, the 
magnet is said to have polarity. The end which points toward 
the north is called its north-seeking pole, and the other end, its 


The reason for this property of the magnet was not known 
until the time of Gilbert. He conceived that the earth is itself a 
great magnet, having its magnetic poles near the geographical 




'poles. To demonstrate that his theory was sound, he made a 
sphere of a piece of lodestone, and showed that a small magnetic 

needle, when near the surface of 
this miniature model of the earth, 
points in directions similar to 
those in which it points at cor- 
responding places on the surface 
of the earth (Fig. 112). 

FiQ. 112. 

Gilbert's Model of the 


200. Magnetic Curves. This 
study of the direction in which 
a small, freely-suspended magnet 
comes to rest in the neighbor- 
hood of a large one, is of great 
interest and importance. The 
experiment may be performed as Gilbert performed it, by placing 
a small suspended magnet at various points near the large magnet, 
and marking at each point the direction in which it comes to rest. 
The same result may be accomplished more quickly by cov- 
ering the magnet with a sheet of cardboard, and sifting fine iron 
filings over it. When the cardboard is lightly tapped, each bit 
of iron acts as a small compass needle, placing itself with its axis 
in the same direction as would a compass needle. Figure 113 is 
a photographic reproduction of the result. It will be noted 
that the iron filings trace well defined curves. These curves 
indicate the direction of the mag- 
netic force at every point about 
the magnet. Many of these 
curves appear to begin at points 
on the magnet near its end. If 
the card were large enough to 
show them all entire, they would 
all appear to end at points on 
the magnet and near its other 
end. There are two points, near 
the ends of the magnet, toward which the lines of force appear 
to converge. They are called the poles. We also note that the 

Fig. 113. 

Magnetic Field of a 
Bar Magnet 



curves do not intersect each other. It is customary to think of 
the lines represented by the filings, as passing through the magnet 
and forming closed curves. 

201. Magnetic Field. Now, the behavior of the iron filings 
tells us that the space about the magnet is permeated with 
magnetic forces, whose directions are indicated by the curves. 
This space around the magnet is called a magnetic 
FIELD. The lines traced by the filings are called lines of mag- 
netic force, because they show the direction in which a free north- 
seeking pole would move at any point in the field. This direction 
that a north-seeking pole would take is often shown by an arrow 

Let us now consider the field of force produced by two mag- 
nets. To do this, we place two magnets under the card, and 
let the iron filings trace 
the directions of the lines 
of force as before. Fig. 
114 shows the result 
when the adjacent poles 
of the magnets are of 
opposite kinds. It will 
be noted that the lines 
of force about each 
magnet are distorted, and that some of the magnetic lines appear 
to pass through both magnets. Since experiment shows that 
unlike magnetic poles always attract each other, we are led to 
conclude that the magnetic force is a kind of tension along the 
lines of force, as if these lines were elastic and were trying to 
shorten themselves. 

If we investigate the shapes of the lines when like poles are 
placed near together, we obtain the curves shown in Fig. 115. In 
this case it will be noted that, though the field of each magnet 
is distorted by that of the other, yet none of the lines of either 
magnet enter the other. Since experiment shows that like 
magnetic poles always repel each other, and since, in the space 
between the two magnets, the lines proceed parallel to each 

Fig. 114. Unlike Poles 



Fig. 115. Like Poles 

other and in the same direction, we are led to think of the lines 

of force as repelling each other in a direction at right angles to 

their lengths. 

We shall therefore 
adopt the hypothesis 
that a magnet is, in 
some way, able to pro- 
duce a strain in the 
medium about it, that 
this strain has a definite 
direction at every point, 

and consists oj a tension in the direction of the lines of force, and 

a repulsion at right angles to that direction. 

202. Permeability. The diagrams with the iron filings 
enable us to foi-m a picture of the attraction between a magnet 
and a piece of iron. For when we place a piece of soft iron be- 
tween two unlike magnetic poles, the shape of the field of force 
is that shown in Fig. 116. It will be noted that the field is dis- 
torted, and the lines of force are concentrated by the iron. Some 
of the lines of force that pass from one into the other go 
through the iron; and so we have attraction. The iron thus acts 
like a magnet, and so, in fact, it is. Its magnetism is said to 
be indv^ced. So we see that iron, when placed in a magnetic field, 
becomes itself magnetic by induction. 

Further, the fact that the lines are gathered in and led through 
the iron shows that the 
magnetic force acts more 
easily through iron than 
through the air. This 
property of gathering in 
and conducting the lines 
of force is called per- 
meability. It is of great 
importance in connec- 
tion with all kinds of apparatus in which magnetic force is 
used. By placing iron in the gaps between magnetic poles, the 

Fig. 116. Permeability 



lines are kept from leaking out; and the force, instead of being 
dissipated, is concentrated within a small space. The more 
nearly we can approach to having a closed magnetic circuit 
of iron, the more efficient the apparatus will be. This is clearly 
shown by the fact that a horse-shoe magnet is much more 
powerful if it is fitted with a soft iron bar, or armature, as shown 
in Fig. 117. 

203. The Earth's Magnetism. The study of the direction of 
the lines of magnetic force about the earth is of great importance 
to navigation, for it has been found that the direc- 
tion of the earth's magnetic lines do not coincide 
with the true north and south direction, and 
that the deviation is different in different places. 
This fact was discovered by Columbus on his 
memorable voyage in 1492, and when it became 
known to his sailors, their fear drove them almost 
to mutiny. As the voyage progressed, it was 
found that the needle did not always point in 
the same direction, but varied from the direction 
of the pole star by different amounts; i.e., the 
magnetic meridian does not always coincide with 
the geographic meridian. 

The departure of the needle from a geo- 
graphic meridian at any point is its declina- 
tion. The declination of the needle and the 
variation in its amount in different places are easily ex- 
plained by assuming that the magnetic poles of the earth do not 
coincide with the geographic poles, and that the needle points 
toward the former, not toward the latter. At points east of the 
magnetic pole, the declination is toward the west, and vice versa. 
From a study of the declination, it has been found that the earth's 
north magnetic pole is situated in Boothia Felix, near Hudson 
Bay, in latitude about 70°.5 north, and longitude 97° west. 

The position of the magnetic pole itself is not absolutely con- 
stant, and therefore we have a variation in the declination at 
different times in the same place. 

Fig. 117. Closed 
Magnetic Cir- 


The positions of the magnetic meridians, as well as other facts 
concerning the earth's magnetism, have been determined by 
different governments, at great expense, because the indications 
of the compass to mariners and surveyors would be very inac- 
curate without the corrections made necessary by the magnetic 

204. The Unit Magnetic Pole. We may now inquire how we 
can compare the strengths of two magnets. In order to do this, 
we must adopt a magnetic unit. The definition of the unit pole 
adopted for scientific work is the following: A unit magnetic 
pole is a pole of mich strength that when it is phiced at a distance 
of one centimeter from a like pole of equal strength, the two repel 
each other with the force of one dyne, 

205. Law of Magnetic Force. When the measurements are 
made in the units just mentioned, it is found that the force with 
which two magnetic poles a^t on ea^h other is equal, numerically, to 
the jyroduxit of their strengths divided by the square of the distance 
between them, 

206. The Chief Characteristics of Magnets niay be summed 
up as follows: 1. When freely suspended, a magnet takes a 
definite position, with its axis nearly north-south; 2, like poles 
repel each other, while unlike poles attract; 3, we conceive a 
magnet to be surrounded by a field of force, made up of lines of 
force whose directions are indicated by the curves traced with 
iron filings; 4, there is tension along the lines of force and repul- 
sion at right angles to them; 5, a unit magnetic pole is one of 
such strength that when it is placed at a distance of 1 cm from 
a like pole, the two repel each other with a force of 1 dyne ; 6, a 
piece of iron or steel, placed in a strong magnetic field, becomes 
a magnet by induction; 7, steel is harder to magnetize than soft 
iron, but it retains its magnetism better; 8, when a magnet is 
broken into pieces, each piece is found to be a magnet. 

207. Electric Currents. At the beginning of our study of 
electricity we were led to ask how dynamos and motors work 


but in order to find this out we had first to learn something 
of electricity in the static condition and some of the proper- 
ties of magnetism. We shall not be prepared to understand 
the operation of electric machinery until we have learned a 
few facts about electricity in motion and the relation between 
electric currents and magnets. 

In a general way we are all familiar with electric currents, 
for we know that they are used to operate the telegraph, the tele- 
phone, the electric light, the trolley cars, and even to ring our 
door bells. Electric currents were wholly unknown until the 
beginning of the nineteenth century. 

The first knowledge of current electricity was derived from the 
researches of Alessandro Volta (1745-1828), who was pro- 
fessor of physics at the University of Pavia. Italian medical 
men had been much interested in investigating the effects of the 
electric shock on animal and himian subjects; and Galvani, 
professor of anatomy at Bologna, was experimenting with the 
legs of a frog. He found that when he twisted together the ends 
of two wires made of different metals, and then touched a muscle 
and a nerve of a dead frog's leg with the free ends of the wires, 
the muscle would contract convulsively, and the legs would kick 
as if they had been brought to life. The greatest excitement 
followed this discovery, and the most extravagant hopes were 
entertained. It was thought that electricity possessed the prin- 
ciple of life, and would cure all ills. 

Volta had already done much experimenting in electrostatic 
induction. He believed that the electricity which caused the 
frog's legs to kick was generated at the contact of the two dis- 
similar metals, and not in the frog's leg, as Galvani contended. 
By a series of very interesting experiments he proved that the 
charge could be obtained from two different metals immersed 
in a liquid and that it did not originate in the frog's leg. 

208. The Voltaic Cell. This discovery was announced in 
the year 1800. While seeking still further to increase the electric 
output of his apparatus, Volta invented the simple cell which 
goes by his name, and wliich has been but slightly modified in 


the best forms of modem commercial cells. This cell consists 
of a copper and a zinc plate, each terminated by a wire and placed 
face to face, but not in contact, in a jar of dilute sulphuric acid. 
By joining a large number of such cells in series, i.e., the copper 
of the first to the zinc of the second, the copper of the second 
to the zinc of the third, etc., effects of considerable power were 
obtained. His electroscope showed that the wire attached to 
the copper was positively charged, and that attached to the zinc 
negatively charged. 

In 1800, only a few weeks after Volta had written of his re- 
searches to the Royal Society at London, two members -of that 
society, Carlyle and Nicholson, were experimenting with Volta's 
apparatus, and discovered that if the terminals were placed in 
water containing a little sulphuric acid, the water would be de- 
composed into its constituent gases, oxygen and hydrogen, and 
would be so decomposed continuously. This decomposition of 
a chemical substance by electricity is called electrolysis. We 
shall learn more of voltaic cells and electrolysis in Chapter XIII. 
It interests us just now, because in this way it was first found 
that the voltaic battery can produce, and transfer along a wire, 
a constant supply or current of electricity. Thus electricity 
was at once transformed • from a subject of purely intellectual 
research 4nto a powerful means of investigation in every branch 
of natural science, and a source of energy whose practical uses 
are so numerous and far-reaching that the boldest imaginings 
of that time are surpassed by the realities of the present. 

209. Electromag^etism. Now, although the method of produc- 
ing continuous currents had been discovered, no one had been able 
to prove that there was any relation between such currents and 
magnetism. That such a relation exists had been suspected, and it 
was discovered in the year 1816 by Hans Christian Oersted (1777- 
1851), professor in the university at Copenhagen. Oersted had 
given much study and thought to the voltaic battery and the 
possibilities of proving the long suspected connection between the 
electric and magnetic forces. In a moment of inspiration, while 
lecturing before his class, the idea occurred to him of joining 


the wires from a battery above a suspended magnetic needle, 
the wire being parallel to the needle but not touching it. The 
needle instantly turned on its axis, and set itself at right angles 
to the wire. He reversed the current, and the needle turned in 
the opposite direction. He had shown that an electric current 
possesses magnetic properties, in that it can move a magnet. He in- 
terposed metals, glass, and other materials between the current and 
the magnet, but found that none of them prevented the action 
of the current on the magnet. Later it was learned that iron 
will screen the magnet from the effects of the current, though 
none but magnetic substances, such as iron, have this effect. 

Oersted's great discovery was published in 1820, twenty years 
after Volta's. The new territory which it opened was immediately 
occupied, and other discoveries quickly followed. To the un- 
trained and unthinking mind, Oersted's discovery might seem of 
little importance; but let us see what we may learn from it by the 
scientific method of inquiry. 

210. The Current Has a Magnetic Field. From the fact 
that the needle always takes a definite position with reference 
to the direction of the electric current, 
we have a right to infer that the cur- 
rent has a magnetic field — that it is, in 
fact, a magnet. 

In order to test this inference, let 
us take a wire conveying a strong cur- 
rent, and dip it into a box of iron filings. ^HAs^i^Ao'jfE^Tic^r^ELD^ 
The filings cling to it, and if carefully 

examined, will be seen to cling to one another so as to form a 
number of rings. They do not stand out radially from the wire 
as they do from the poles of a magnet, but are like a lot of 
curtain rings strung along the wire. Break the current, and they 
fall off. The field is instantly destroyed. In order more con- 
veniently to investigate the form and extent of the field, let us 
pass the wire up through a small hole in a smooth board, and 
then down through another hole, close the circuit, sprinkle 
filings, and tap the board. The filings are seen to jump into 







concentric rings around the two portions of the wire (Fig. 118). 
The lines of force, then, are circumferences of concentric circles, 
whose planes are all perpendicular to the direction of the current. 
In which direction will these forces cause a north-seeking 
pole to move? If we place a pocket compass at various points 
around the wire, and mark by short arrows the directions of the 
needle at these points, using the arrow tips to denote the direction 
of the north-seeking pole of the needle, we shall see at once, from 
the two maps, that when we look along the wire in the direction in 
which the current is going; i.e., from copper to zinc, the direc- 
tion of every line of force is that in which the hands of a clock re- 
volve. Where the current is coming up through the board, the 
lines of force appear to circulate counter-clockwise, but that is 
only because we are facing the current so that it is moving 
toward us. If we were to get down under the board and look 
upward, the lines would appear to be clockwise as before. 

211. Test for the Direction of a Current. This rule enables 
us to predict the direction which the needle will take when placed 
near a current, and conversely to tell the direction of the current, 
if unknown, by the direction which the needle takes. Thus, if we 
explore the field around the wire and find that there is a deflection 
of the north-seeking pole clockwise with reference to it, we know 
that there is a current traversing the wire, and that it is flowing 
away from us. Conversely, if the deflection appears counter- 
clockwise to us, then the current must be' coming toward us. 

212. How are Currents Belated to Magnets? Notice again 
the field of the current-bearing loop. Look along the top 
of the loop in the direction of the current. The positions taken 
by a compass needle show that all the lines go out of the left-hand 
face of the loop, and enter the right-hand face, curving around on 
the outside of the loop. Referring back to the field of the bar 
magnet (Fig. 113), does not this suggest that the right-hand face 
of the loop is a north-seeking pole? Make another loop with a 
smaller diameter and with the lines closer together, the field 
smaller and stronger. 



■'*"'■ '1, ■ \ 


. ^^ 

.:- • ' '^. ' 

/ - 

■ ■■■ - X ■, 




■ '^% 



Fig. 119. 

Magnetic Field of 
A Coil 

Can we not increase the strength of the field by making a coil 
of several loops, so as to get the lines of all the loops into the same 
space? On trying this, we find the 
needle and filings more strongly 
affected, and the map. Fig. 119, 
much more distinctly like that 
which a magnet ought to give us, 
if it were made short and very 
thick compared to its length. 

Follow up this suggestion by 
making a new coil in the form 
of an elongated helix, and more like the bar magnet in shape 
(Fig. 120). The field of this helix looks almost exactly like that 
of the bar magnet. Each little loop has its own small circular 
lines, but inside and outside the helix they combine to form 
strong resultant lines which are closed curves exactly like those 
of the magnetized steel bar. 

We now have a right to infer that our current-bearing coil, 
or helix is a veritable magnet. Will it do the things that a magnet 
does? We have seen that it strongly attracts iron, and that like 
the whole coil every little loop does so, though less strongly. 
But, if freely suspended, will it point north and south? If placed 
near another suspended magnet or another current-bearing loop 

or helix, will the poles mu- 
tually attract? Are the effects 
strongest at its poles? The 
requisite experiments show 
that in every case the answer 
is "yes," a current-bearing 

conductor is a magnet, 

213. Permeability— Elec- 
tromagnets. We found (Art. 
202) that soft iron placed in 
the field of a bar magnet, 
gathers in the magnetic lines, i.e., concentrates the force. Ought 
we not, then, to be able to increase greatly the force of the coil or 

Fig. 120. Magnetic Field of a Helix 



FiQ. 121. An Iron Core Strength- 
ens THE Field 

the helix by putting soft iron into it? On trying the experiment, it 
is found that the iron core adds immensely to the magnetic strength 

of the coil or the helix. We have 
thus been led to construct an elec- 
tromagnet (Figs. 121, 122). Such a 
combination has two great advan- 
tages over a steel magnet: 1. 
Since its strength is proportional 
to the strength of the current, and 
also to the number of turns of the 
wire aroimd the coil, we can thus 
make a very strong magnet. 2. We can magnetize it and de- 
magnetize it at will. We may now ask, can it not be maide to 
do Various kinds of mechanical work at some distance from the 
point at which the electric circuit is opened and closed? 

This question leads us into a field in which notable discoveries 
were made by an eminent American, Joseph Henry (1799-1878). 
Henry was a busy teacher in the Albany Academy, and after- 
wards professor at Princeton, and Secretary of the Smithsonian 
Institution at Washington. He was incessantly overworked, but 
in spite of that fact he made 
researches that brought him 
high honors among, scientists 
all over the world. 

214. The Electromagnet 
and the Telegraph. Pop- 
ularly, Henry's- name is 
scarcely known in connection 

Fig. 122. Field of an Electromagnet 

invention has been given by 
the American public to Sam- 
uel F. B. Morse. But what Morse did was simply to combine prin- 
ciples and apparatus discovered by Henry and others, and make 
the public believe in the possibilities of the electromagnetic tele- 
graph. Important as were the services of Morse, the honor of the 


invention belongs to Henry, who, like his great English friend Far- 
aday, was content to make fundamental scientific discoveries and 
leave their practical applications to others. Recognizing the pos- 
sibilities offered by the electromagnet, physicists everywhere were 
trying to construct an apparatus for signaling at a distance by 
means of it, and had given it up, because the current became so 
weak after traversing a few hundred feet of wire, that the magnet 
would not move anything. A member of the Royal Society had 
tried, and claimed that he had demonstrated the impossibility of 
the scheme. Henry read this paper and to his mind it was a 
challenge; so with characteristic American audacity he set out 
to accomplish the impossible. 

Henry discovered that if he insulated his wires by covering 
them with silk, and then woimd many turns of fine wire on the 
core as thread is wound on a spool, this long coil magnet 
would work at great distances from the battery, even though the 
current was very weak. Using the horse-shoe form of core, he 
pivoted in front of it a lever, carrying a little soft iron bar 
or armature. The lever terminated in a clapper, which would 
strike a bell when the armature was attracted. Placing this 
apparatus in circuit with a battery of many cells in series, 
he was able to make the clapper strike the bell whenever he 
closed the circuit, and fall back in obedience to the tension of a 
spring whenever he destroyed the magnetic field by breaking the 

Thus, representing each of the letters of the alphabet by a 
combination of bell strokes, a message could be spelled out. In 
the instrument which Morse afterwards patented, the lever car- 
ried a pencil which it pressed against a moving roll of paper when 
the armature was attracted. By making short and long contacts, 
dots and dashes were made on the paper strip; and in the 
Morse Alphabet each letter was 'represented by a combination of 
dots, dashes, or spaces, thus: 

a g t m 1 




The original idea of the Morse recorder survives in the "tick- 
er*' of the stock exchanges and brokerage offices to-day, but the 
instrument now generally used for commercial telegraphy is the 
speedy and more familiar *' sounder," Fig. 123. Each signal of 

the sounder is begun 
by the fever L clicking 
against a stop P when 
the armature A is 
pulled down on closing 
the electric circuit, 
which passes around 
the core of the magnet 
M. It is ended by the 
lever clicking against 
another stop Q when 
the circuit is opened, and the armature, released from the magnetic 
attraction, is pulled back by a spring 8. Thus the differences 
between the dots and the dashes are represented by differences 
in the time intervals between the double clicks of the sounder. 

Fig. 123. Telegraph Sounder 

215. The Belay. Neither a bell nor a sounder will work 
over lines many miles long, however, because the current, weak- 
ened by the resistance of the long circuit of wire, can not pro- 
duce enough energy to do the heavy work. Henry overcame 
this difficulty by devising the relay. He found that a single 
battery cell with large plates would produce great effects when 
the circuit was of short, thick wire,, offering little resistance to 
the current, and that powerful magnetic effects could be produced 
with such a battery by using a few turns of thick wire around the 

Suppose we have such a battery and magnet at the receiving 
station connected in a circuit by 'short, thick wires. This magnet 
SM, Fig. 124, with a strong current from a local battery LB can 
operate the sounder lever SL, Now let us place in the main 
LINE CIRCUIT L a magnet RM wound with many turns of fine wire. 
The main current, though exceedingly weak, can furnish enough 
energy to this sensitive magnet, so that it can work a very 



light armature A, Fig. 125, with its lever. Now, this light 
lever of itself can neither do any printing nor make any noise, 
but it may easily make or br^ak an electrical contact between 
two little points CC, one of which projects from the lever and 
the other from a fixed metal post. 

Suppose we cut the wires of the local circuit, and join 
one of the cut ends to the contact point C on the relay lever, and 
the other to the fixed point C against which^it strikes, as in Fig. 124. 
All we have to do now to work the big 
sounder, by means of the key at the 
distant station, is to send our weak 
main line current L around the core 
of the sensitive relay magnet RM, The 
relay armature is instantly attracted 
and the contact point on the light ar- 
mature lever strikes the fixed contact 
point C. This completes the local cir- 
cuit, and lets the powerful local current 
gq around the core of the sounder mag- 
net. The sounder lever is drawn down 
and makes a loud click. Open the 
main line circuit; and the relay arma- 
ture, in obedience to the tension of its 
spring S, Fig. 125, flies back. This 
separates the contact points CC, thus 
opening the local circuit; and the 
local current ceases to flow around the 
sounder magnet. Instantly, in obedi- 
ence to the tension of its spring, the 
sounder lever flies back. Thus the powerful local current is released 
or throttled at will by the operator at the distant sending station. 

This was the principle of Henry's relay. It is something like 
using the weak current to pull a hair trigger, and discharge a 
big gun. We make a powerful source of energy do heavy work, 
but we control it by a weak current from a distant source. Long 
distance transmission of electric signals of any kind is commer- 
cially a practical impossibility without the relay. 

Fig. 124. Telegraph Diagram, 
One Station 



Fig. 125. Telegraph Relay 

216. The Telegraph Key. The opening and closing of the 
circuit is accompHshed by a key, Fig. 126 and K, Fig. 124, worked 

by the thumb and first 
C C 

-^^^ two fingers of the ope- 

rator. It is simply a 
lever with a contact 
point P attached to its 
under side, and strik- 
ing another . contact 
point attached to the 
metal base, but insu- 
lated from it. In tel- 
egraphic practice there is a key, relay, sounder, and local bat- 
tery at each station; and when one key is worked, it operates all 
the keys and their sounders simultaneously. Accordingly, each 
station has its own signal letter or call by which the attention 
of its operator may be attracted. 

217. Grounding the Wires. Before practical telegraphy had 
progressed very far, the fact that the earth conducts electricity 
suggested the possibility of using the earth as part of a telegraphic 
circuit. This idea is realized in the following manner: One of 
the battery wires is joined to a metal plate or pipe G, Fig. 124, 
buried in damp ground, and the farther end of the line wire is 
grounded in the same manner. The current which has passed 
from the other pole of the battery along the line wire and through 
all the instruments, proceeds to the ground connection at the 
farther end, and com- 
pletes its circuit 
through the earth. 
It is somewhat as if 
we had a pump on 
one side of a lake, 
which would lift 
water and force it 
through a pipe to a place on the other side of the lake, where 
it might do work in turning water motors, and then return to 

Fig. 126. Telegraph Key 



the lake when its energy was exhausted. Just as in the case of 
the water there is a current through the pipe, and a drift across 
the lake, so in the case of the grounded electric current, there 
is an electric current in the wire, and an electric drift back along 
the ground. As the ground resistance to the electric drift is very 
small, this arrangement not only saves copper, but also saves en- 
ergy, as we shall see later on. 

218. The Electric Call Bell. The electric call bells and buzz- 
ers in common use work very much like a telegraph sounder, 

except that the armature which carries the ^ 

bell clapper, or the reed which makes the 
tone of the buzzer, is made to vibrate auto- 
matically, as long as the current is con- 
tinuously supplied to it by keeping a push 
button depressed. 

The construction and operation of the 
bell will be understood from the diagram, 
Fig. 127. The path of the current may be 
traced by the arrows. When the button is 
pressed, the circuit is closed, the armature 
is attracted by the electromagnet, and the 
clapp_er strikes the bell; but at the instant 
when this happens, the contact breaks be- 
tween the contact-screw C and the armature, 
because the armature has been pulled away from the contact 
screw. The breaking of the electric circuit at this point destroys 
the magnetic field; and the armature, no longer attracted, is 
pulled back by the elastic spring on which it is mounted. Thus 
it touches the contact screw; and since the circuit is now again 
closed, the operation is repeated. Thus the armature vibrates 
automatically, receiving its periodic impulses from the periodic 
magnetic field. 

Fig. 127. The Electric 

219. Galvanometers. Shortly after Oersted's discovery. Am- 
pere suspended a delicate magnetic needle inside a coil of wire, 
and used it for detecting the presence of a current and estimating 



its intensity. Such an instrument is called a galvanoscope or a 
galvanometer. When the plane of the coil is placed in the mag- 
netic meridian, the needle, directed by the earth's magnetism, 
remains in the plane of the coil — ^i.e., with its axis north-south — 

unless a current is passed around 
the coil. In this case, the magnetic 
field of the current exerts a moment 
of force on the needle, tending ,to 
pull it around so that it will lie 
along the lines of force of the coil — 
i.e., east-west; and the angle through 
which the needle is turned from the 
north-south direction depends upon 
the strength or intensity of the cur- 

Many different types of the galva- 
nometer are now in use. The most 
useful are: (1) The astatic, a gal- 
vanometer which may be made so 
sensitive that it can be used for re- 
ceiving signals sent through the sub- 
marine cables, (2) The tangent gal- 
vanometer, commonly used in measuring currents of considerable 
strength, and (3) The D'Arsonval galvanometer. This last form 
we shall pause to examine, both because it is the best for all- 
around use, and because it will assist us in making another im- 
portant step in our inquiries. 

In the other forms we have a fixed coil with a needle sus- 
pended at its center. In the D'Arsonval (a simple form is shown 
in Fig. 128) we have a large, fixed, U-shaped steel magnet, with 
a small coil suspended between its poles. The current to be 
tested enters at the binding post E-\-, passes into the coil through 
a thin metallic ribbon R by which the coil is suspended, and passes 
out below by a similar ribbon. When the current passes 
through the coil, the latter becomes a magnet and tends to set 
its lines of force in the same direction with those of the large 
fixed magnet. The magnetic lines of the magnet and of the coil 

Fig. 128. D'Arsonval Galva- 


are represented by arrows. The deflections of the coil are read 
by a pointer moving over a circular scale, or by the displace- 
ment of a beam of light, reflected from a mirror attached to the 
coil. With all galvanometers the deflection of the suspended coil 
or magnetic needle is greater when the current is greater. The 
exact relation between the deflection and the strength of the current 
depends on the kind of galvanometer. 

}. A Suggestive Experiment. Let us pass the current from 
a single voltaic cell through the coil of D'ArsonvaFs combinatioh. 
The coil, Fig. 128, hangs with its plane parallel to that of the 
U-magnet. As soon as the current passes, one face of the coil 
becomes a north-seeking pole and the other a south-seeking pole, 
as is suggested by the short, curved arrows which represent two 
of its lines of force. The like poles of the magnet and coil repel 
each other and their unlike poles attract each other. So the coil 
turns through a right angle, and stops with its north-seeking 
face next the south-seeking pole of the magnet. 

If we reverse the current in the coil, its magnetic poles will 
be reversed also; so that each pole of the coil will be facing a like 
pole of the magnet. Now, we know that like poles repel each 
other and unlike poles attract; therefore we may infer that the 
magnetic forces will cause the coil to face about, so that each of 
its poles will be adjacent to an unlike pole of the magnet. 
When we shift' the battery wires, so as to reverse the current 
through the coil, we find that the coil turns through half a revo- 
lution, just as we inferred. 

With the exercise of the scientific imagination we may now 
arrive with a single bound at the principle of one of the greatest 
inventions of all time. Can we, by any device, modify this appa- 
ratus, so that when the current is again reversed, so as to be in 
its original direction, the coil will turn through the second half 
of the circle, instead of going back on its path? In other words, 
can we, by reversing the current at the end of a half turn, cause 
the coil to rotate continuously, instead of merely vibrating back 
and forth through a semicircle? If we can but do this, what 
boundless possibilities wait on the labors of inventors! Electric 


motors, turning machinery miles away from the source of current 
— electric power, distributed by wires everywhere and converted 
into mechanical work in the factory, the street or even in the private 
home — power in just the amount wanted, available at the instant 
when it is wanted, and the expense stopped the instant the power 
is not needed, by merely turning a switch; these are among the 
achievements which the device suggested would make possible. 

This was the dream that possessed the imaginations of scientists 
at the beginning of the second quarter of the nineteenth century. 
Before the middle of that century the necessary discoveries had 
all been made. The principles were in the hands of the inventors, 
and in another forty years thousands of motors were in successful 

221. The Electric Motor. In our electrical studies thus 
far we have learned some of the most important facts and 
principles that were known when Faraday and Henry were 
seeking to discover the principles of the electric motor. Suppose 
that the motor exists only in our imaginations as it then did in 
theirs, and returning to our magnet and little suspended coil, let 
us see what we can discover. 

Let us send the current into the coil and observe what happens. 
The coil rotates through a right angle, and sets its faces opposite 
the poles of the magnet; but it does not stop at the instant when 
it reaches that position. On account of its inertia, it goes 
beyond that position. The magnetic force and the torsional elas- 
ticity (twisting force) of the suspending ribbon stop it and bring 
it back. After a few oscillations it settles into the position just 
mentioned, in which its lines of force coincide with those of the 

If we can manage to reverse the current just as it passes this 
position, and if we can also free the coil from the torsion of the 
suspending ribbon, then instead of oscillating and settling in the 
definite position mentioned, it will go on around through half 
a revolution more. But on account of its inertia, it can not stop 
itself; and if we again reverse the current just at the right instant, it 
will continue to rotate through another half turn. Thus it appears 



that if we can reverse the current just at the end of each half turn, 
and if we can also get rid of the twisting force of the suspension, 
we may produce continuous rotation. 

By a little practice in timing the reversals of the current, we 
may easily make the coil of a galvanometer execute one or two 
complete revolutions, for in this case it stops only when the sus- 
pension ribbon gets twisted up. Evidently we must overcome 
this mechanical difficulty. This may be done by inventing a 
sliding contact. 

Armature, Field Magnets, Collecting Eings, Brushes. 
We can arrange such a sliding contact by fastening the coil to 
a steel axis or shaft, whose ends turn freely in suitable bearings. 
It will be better also to mount the shaft horizontally rather than 
vertically (Fig. 129). We shall now call the coil an armature and 
the U-magnet the field 
MAGNET. Then we may 
place near one end of the 
axis a pair of metal rings, 
and solder the ends of the 
coiled wire to the rings. 
These rings, R -f and 
R —, we shall call the 


order to let the current in 

at one of these collecting rings and out at the other, we attach the 
battery wires to two light metal springs, B + and B — , which we will 
call the BRUSHES, and hold these brushes so that one of them shall 
rub lightly against each of the collecting rings. While making the 
apparatus, there is something very important which we must 
remember and provide against. The steel shaft, or axis, about 
which the coil rotates, is a conductor; and if our collecting rings 
were in metallic contact with it, the greater part of the current, 
coming to the first ring, woiild take the short cut along the steel 
shaft from this ring to the other, and thence back to the battery. 
Only a very small fraction of it would go around the turns of the 
coil. Therefore our collecting rings must be mounted on a sleeve 

Fig. 129. The Coil is Fkee to Rotate 



of hard rubber which will insulate them from the armature shaft, 
and from each other. 

Having ready this apparatus, we make the experiment — first 
touching the brush B -\- to ring R +, and the brush, B — to 
R— ; then reversing the brushes, so that the brush B + touches R — 
and the brush B — touches JB-f. If we are skillful enough in 
shifting the brushes at the right instant, we secure continuous 
rotation. We have, then, experimentally demonstrated the possibil- 
ity of the electric motor; but our apparatus is as yet very crude 
and inefficient^ even for a toy. What we need now is either an 
alternating current, or a device for making the coil itself reverse 
the current from the battery. We shall not here consider 
the first proposition, because when the invention had reached 
this point nobody had even dreamed of producing an alternating 
current; i.e., one that would flow first in one direction and then in 
the other. 

223. The Commutator. Since tKe necessary motion of shift- 
ing is only relative, it makes no difference whether the brushes 

reverse their positions 
while the positions of the 
collectors remain fixed, 
or whether we attach the 
brushes to a fixed insu- 
lated support, and let 
the ^ collectors reverse 
their positions. How 
can we do this? 

A little imagination 
and some careful thinking will help us to hit on a plan. Man- 
ifestly, when the coil turns, one-half of the ring is passing you 
during the first half of the rotation, and the other half of the ring 
is passing you during the other half of the rotation. It is, then, 
easy to solve the problem by discarding one of the rings, split- 
ting the other ring parallel to the shaft XX, and attaching the 
two ends of the coiled wire to the two separated halves of the 

Fio. 130. The Motor Diagram 


We must now fix the brushes so that they touch diametrically 
opposite points of the split ring. This arrangement is shown in 
Fig. 130, the shaft being broken away and the split ring (C +,C —) 
being enlarged for the sake of clearness. 

If the brushes are properiy placed, it will be seen that at the 
instant when the armature comes into the position where its poles 
(i.e., its flat faces) are nearest the attracting poles of the field 
magnet, the half ring C + will shift from the brush B + to the 
brush B—, and the half ring C— will shift from the brush B— 
to the brush B +. The result is that the current through the 
armature coil is reversed, and so. its poles are also reversed. 
Therefore the armature poles are repelled by the poles of the £eld 
magnet, and so the coil continues to rotate. Since a similar rever- 
sal of the current will occur at the completion of every half turn 
of the armature, the latter will continue to turn so long as the 
necessary power is supplied. 

This split ring device we call a commutator, because it re- 
verses the current through the coil whenever it shifts the brush 

The problem, then, of providing a sliding contact, and a 
device for automatically reversing the current through the arma- 
ture, is satisfactorily solved in our imaginations. If now we have 
the completed apparatus before us, and send into the armature a 
current of sufficient power, we shall be rewarded for all our labors 
by seeing the armature spin merrily round and round. The solu- 
tion of the problem is now a reality. 

224. From Toy to Practical Machine. We must not stop 
at this point, however, for our motor is very weak, being barely 
able to run itself, to say nothing of driving machinery; and, fur- 
thermore, we already possess knowledge which we may apply 
to increase its efficiency. We know that the turning force is that 
of the two magnets, and we have learned some ways of making 
these magnets stronger. 

1. We know of the permeability of soft iron (Art. 202), so 
the next step is obvious. It is to fill the coil with a soft iron 



2. We know that an electromagnet will have a stronger field; 
so we can further increase the strength of the field, by substituting 

an electromagnet for the 
steel niagnet as in Fig. 131. 

3. We may make our 
magnet shorter and thicker, 
with large pole-pieces, shaped 
so as to embrace the coil as 
closely and completely as 
possible without interfering 
with its rotation (Fig. 131). 
This fills the air-gaps as 
nearly as may be with soft 
iron, and gathers in the lines 
of force; so that more of 
them pass through the arma- 
ture, instead of leaking away 
where they will not be util- 
By a little further modification in shape, we may make the 
completely **iron- 

FiG. 131. Motor Frame, Poles, and Field 


clad," as, represented in 
Fig. 132. This is one of 
the modern forms and 
leaves little to be desired 
in the matter of packing 
the lines of force into the 
effective space, and pro- 
tecting the machine from 
anything that might get 
into it. 

4. We may greatly in- 
crease the efiPectiveness of 
the armature by winding 
on the core another coil at 
right angles to the first, 
so that when one coil is turning into the least effective position, the 
other is turning into the most effective position. The number of 

Fig. 132. A Modern Motor 



Fig. 133. Motor Armature, Commutator, 
AND Shaft 

coils may be increased to four, six, eight, and so on (Fig. 133), 
till all the available space on the core is filled, but the commutator 
must be then re-di\dded so as to have one segment for each 
coil. Increasing the number of coils not only increases the 
magnitude of the force, 
but also makes it approx- 
imately uniform in inten- 

225. Winding the Field 
Magnets. The current may 
be passed by the brushes 
through the armature coils and thence through the field coils 
(series winding, Fig. 134), or it may diyide at the brushes, part 
going through the field coils and the remainder through the 
armature coils (shunt winding, Fig. 135), or both these methods of 
winding may be used together (compound winding, Fig. 136). 
There are many other ways employed in the construction of 

modern motors, by 
which their efficiency is 
still further increased. 
Figs. 131 to 133 are 
photographs of one of 
the best modem types 
of direct current motor. 

226. Ampere's The- 
ory of Magnetism. A 
theory of magnetism 
which was proposed by 
Ampere is generally 
held at present, though 
it has received consid- 
erable extensions in order that it may describe some facts discov- 
ered since Ampere's time. 

If we break up a magnetized knitting needle, we find that each 
part is a magnet, even to the smallest parts into which it may be 

II * 

I" 'f 

Fig. 134. Four Pole Motor, Series Wound 



Fio. 135. Two-Pole Motor, 
Shunt Wound 

broken. There is no reason why we should believe that the 
smallest particles of iron into which a magnet can be divided, 
should fail to have their magnetic poles. Ampere, therefore, be- 
lieved that the magnetic 
properties were permanently 
possessed by the molecules 
of the iron. The amended 
and extended hypothesis of Ampere is: 

1. Every molecule of a magnetic 
substance is a magnet, possessing poles, 
and all the properties with which we 
have become familiar in large magnets. 

2. When the magnetic substance is 
unmagnetized, the magnetic axes of the 
molecules are turned in all possible di- 
rections, so that their magnetic forfces neutralize one another. 

3. When the lines of force of an external magnetic field pass 
through the bar, the little molecular magnets tend to set them- 
selves along the lines of the external field. 

4. When the molecules are all as nearly parallel to the axis of 
the bar as is possible, the bar is said to be magnetically saturated. 

5. The perfect magnetization of a molecule may be ac- 
counted for by supposing, either that an electric current is always 

circulating around it in a plane per- - _ 

pendicular to its axis, or that it has 
an electrostatic charge and is rapidly 
spinning. In view of the experiment 
described in the next article the latter 
idea seems to have the advantage. 

227. Magnetic Field of Moving 
Charges. In order to find out whether 
a rapidly moving charged particle has 
a magnetic field, just as a current 
has, Rowland devised the following experiment: A metal disc 
was mounted so that it could be rotated about an axle perpen- 
dicular to its plane. When this disc is charged electrostatically, 

Fig. 136. Two-Pole Motor, 
Compound Wound 


and is also rotating rapidly, it is found that a compass needle 
near the disc is deflected just as it would be if a current of elec- 
tricity were flowing in the path -of the rotating charge. Thus we 
learn that charged bodies in rapid motion froduce electromagnetic 
effects, just as currents do; and therefore we are led to conceive 
that a current may be simply a series of electrostatically charged 
particles darting rapidly along a wire. 

The numerical relations obtained from this experiment are very 
suggestive. It was found that if a unit charge (Art. 191) moves 
with a velocity of 3 X 10*° ~, the strength of the magnetic field 
produced is the same as that of a unit magnetic pole (Art. 204). 
We shall leam in Chapter XXII that both light and the electric 
waves, predicted by Maxwell and detected by Hertz (Art. 198), 
travel with this same velocity, 3 X 10*® ^. These facts suggest 
the idea that there must be some intimate relation between elec- 
tricity and light. In Chapter VIII we learned that heat and light 
are related phenomena. Therefore we may reasonably ask whether 
electricity, heat, and light may not be simply different manifesta- 
tions of one and the same form of energy. This question will be 
considered further in the later chapters. 

228. Energy of a Magnetic System. A magnet can not do 
work, unless mechanical work is done on it, or a current of 
electricity gives up some of its energy to it. 

Thus, if a magnet has potential energy, so that it can attract 
a piece of iron to itself, work had to be done to store up this energy. 
Either the iron had first to be pulled away from the magnet, or the 
magnet had to be pulled away from some other magnet from 
which it got its magnetism. 

If an electromagnet does work in moving a piece of iron, some 
of the current which energizes the magnet is used up in doing this 
work. Thus, if we pass a current through an electric lamp and a 
magnet coil, the lamp diminishes in brightness while the work is 
being done by the magnet. If we replace the lamp by a gal- 
vanometer, the deflection diminishes while the magnet is doing 
its work. 

This is only another particular case of the general law of the 


CONSERVATION OF ENERGY, which states that no energy can be 
created or destroyed. 


1. Lodestone is a compound of oxygen and iron, and is a nat- 
ural magnet. 

2. A magnet attracts iron, and when freely suspended, takes a 
definite position with its axis nearly north-south. 

3. The magnetic properties of a magnet are strongest at its 
ends, which are called its poles. 

4. Like magnetic poles repel each other, and unlike poles 
attract each other. 

5. A magnet induces magnetic properties in every piece of 
iron or other magnetic substance which is brought into its neigh- 

6. Every piece of a magnet is itself a magnet. 

7. The poles of a magnet may be determined, if unknown, 
either by suspending it freely, or by presenting its poles to those 
of a magnet. 

8. The space that surrounds a magnet is a field of magnetic 

9. A vector drawn tangent to a line of force shows the direc- 
tion in which a free north-seeking pole is urged at the point of 

10. Lines of force are always closed curves which pass out- 
side the magnet from its north-seeking to its south-seeking 
pole, and inside the magnet from its south-seeking to its north- 
seeking pole. These lines never intersect each other. 

11. Magnetic forces act as if the lines of force were elastic 
threads which tend to shorten themselves lengthwise and repel 
each other side wise. 

12. If soft iron is placed in a magnetic field, the lines of force 
are gathered in and pass through it. This is due to permeability 
of the iron. 

13. The earth is a great magnet, having one of its magnetic 
poles near Hudson Bay. This explains why a magnetic needle 
points nearly north-south. 


14. The declination of the needle is explained by the fact that 
the earth's magnetic poles do not coincide with its geographical 

15. Variations in declination at a given place are caused by 
the small variations in the positions of the earth's magnetic poles. 

16. A unit magnetic pole is one which, when distant 1 cm 
from an equal like pole, repels it with a force of 1 dyne. 

17. The force between two magnetic poles, expressed in dynes, 
is equal to the product of their magnetic strengths divided by the 
square of the distance between them. 

18. By chemical action a voltaic cell can supply a continuous 
current of electricity, which transfers energy along a conductor. 

19. An electric current has a magnetic field, and can do me- 
chanical work by moving a magneti 

20. Only magnetic substances act as screens to cut off 
magnetic force. 

21. The magnetic lines of a current in a straight wire are 
circumferences of circles, whose planes are perpendicular to the 
direction of the current. 

22. The direction of a line of force is always clockwise to an 
observer looking in the direction in which the current is going. 

23. The magnetic field of a current-bearing helix is similar 
to that of a similarly shaped steel magnet. 

24. This fact leads us to infer that a current-bearing helix will 
always behave like a magnet, and this conclusion has been fully 
verified by experiment. 

25. Placing a soft iron core in a current-bearing helix increases 
the strength of its field. Such a combination is an electromagnet. 

26. Electromagnets are used in telegraph sounders and call 
bells to send signals over short conducting lines, and in relays 
to send them over long lines. 

27. Galvanoscopes are used to detect currents and determine 
their directions; galvanometers, to measure their intensities. 

28. An electromagnet, mounted on an axis and placed between 
the poles of another magnet, may be made to rotate continuously. 
This is the principle of the electric motor. 

29. The efficiency of an electric motor may be greatly increased 


by arranging the soft iron parts so as to form a closed magnetic 
circuit, apd also by increasing the number of coils in the armature. 

30. According to Ampere's theory of magnetism, every mole- 
cule of a magnetic substance is supposed to be itself a magnet, 
because it carries an electric charge and constantly spins on its 

31. A unit electrostatic charge, moving with a velocity of 
3 X 10*® ^, is equivalent to a unit current and therefore produces 
a unit magnetic pole. 

32. A magnetic system can do no work unless it has been sup- 
plied with energy, either by mechanical work, or by using up 
some of the energy of an electric current. 


1. Describe a series of simple experimeuts for demonstrating the 
properties of magnets. 

2. Explain how you can prove whether a substance is strongly 
magnetic or not. If it is a magnetic substance how can you prove that 
it is or that it is not a magnet? 

3. What is a magnetic field? What is a line of magnetic force? 

4. Describe the methods of mapping a magnetic field. 

5. Sketch the appearance of the magnetic field of a bar magnet; 
of two like poles repelling each other and of two unlike poles attracting 
each other. 

6. Sketch the effect on the field of introducing soft iron into the 
magnetic circuit. What name is given to this property? What are 
the effects on the properties of the field?, 

7. What is the magnetic meridian of a place? 

8. What great principle was established by Oersted's discovery? 

9. Describe experiments by means of which the magnetic field of 
a current can be shown. 

10. How may we determine the direction of a current by means 
of a magnetic needle? 

11. Sketch the field of a current-bearing loop. How is the mag- 
netic strength affected by multiplying the number of turns of wire so 
as to make a compact coil? 

12. What general fact may be inferred by the resemblance between 
the field of any closed electric circuit and that of a magnet? 

13. Suggest a series of experiments by which the fact thus inferred 
may be verified. 


14. What is an electromagnet? What advantages has it over a 
permanent magnet? . ^ 

15. Diagram an electric call bell, trace the current, and explain 
its action. 

16. What is the principle of the galvanometer? State its uses. 

17. Diagram a D' Arson val galvanometer and explain its action. 

18. Describe the modifications that will convert D'Arsonval's 
combination into an apparatus producing continuous rotation. 

19. What are some of the most important changes that must be 
made in the details of this apparatus in order to make an efficient 
electric motor? 

20. Describe the experiment of the rotating charged disc and tell 
what conclusions we may draw from it. 

21. Show that a magnetic system can have energy and do work 
■only when energy has been supplied to it. 


1. Write the expression that represents the magnitude of the force 
/ in dynes, between a north-seeking magnetic pole of strength s and a 
south-seeking pole of strength s' placed d cm apart. Is this force 
attractive or repulsive? 

2. When two south-seeking magnetic poles of 2 and 3 units strength, 
respectively, are placed 6 cm apart, with what force do they affect 
each other? Is the force attractive or repulsive? 

3. What is the strength of a magnetic pole which exerts an at- 
traction of 1 ,000 dynes on another pole which is distant 20 cm and has 
a strength of 25 units? 

4. With the aid of outline diagrams, describe the construction 
and operation of the telegraphic key, the sounder, and the relay. 

5. Diagram a telegraphic circuit of two stations, without relays, 
trace the current by arrows, and explain the operation of sending a 

6. Diagram a complete telegraph circuit for two stations, with 
ground connections, battery, keys, relays, local circuits, local batteries, 
and sounders. Trace the main line circuit around With black-ink 
arrows, and the local circuits with red-ink arrows. Explain the action 
of all the instruments when a key is worked. 

7. Remove the cover from the push button of your electric door 
bell, examine the contact spring carefully to find out how it works, 
and make a diagram explaining its action. 

8. Diagram a door bell battery of two cells in series, as you will 
find them connected (c/. Art. 218). To this diagram add one of the 
bell, and connect the battery with the bell by a wire. Add to the dia- 
gram a wire returning from the bell through a push button to the 


battery, so as to make a complete circuit when the button is pressed. 
Trace the current through the circuit by means of arrows, and explain 
the action. 

9. To your bell diagram, add several push buttons, as they would 
be used to ring the bell from different points of the house. 

10. Make a diagram representing a bell circuit in which the current 
from the battery divides among several bells and reunites in a single 
wire which conducts it back to the battery. Put into the diagram 
one push button by which all the bells may be rung together. 

11. Diagram an arrangement in which five bells are placed in a row, 
and a current may go from a battery through any one of the he\\», 
from thence through a push button in a distant room, and from the 
push button to. a common return wire which conducts it to the battery. 
This is the arrangement of the annunciator seen in the offices of 
hotels. There is a push button and an indicator for each room. 


1. With a ten-cent toy magnet, a few sewing needles, knitting nee- 
dles, some bits of broken watch springs, which your jeweler will give 
you, a few bits of cork, some sealing wax, and a small amount of in- 
genuity, you may verify all the properties of magnets, mentioned in 
Arts. 199-206. 

2. If you have a little shop of your own with a few good tools, 
you may easily make yourself a simple D* Arson val galvanometer, 
like that shown hi Fig. 128, a working telegraph set, a small motor, 

3. You will find much helpful information about making such 
things in Hopkins's Experimental Science; Electric Toy Making^ by 
T. O'Connor Sloan (Norman W. Henley & Co., New York), and in a 
series of little handbooks of the Bubier Publishing Company, Lynn, 

4. Get the necessary information from an electrical supply store, 
and take charge of the electric bells in your home, keeping the battery 
in order and the bells in adjustment. 

5. Examine the field windings of a toy motor. Is it shunt wound 
or series wound? See if you can change the connections so as to 
convert it from one style of winding to the other. A shunt winding 
requires many turns of fine wire, a series winding few turns of coarser 

6. Connect a toy motor with a battery and a galvanometer, and 
note the deflection. Now hold the armature so it can not rotate, and 
see if the deflection is greater. If the first deflection is too great 
connect a shunt across the galvanometer terminals (c/. Art. 268, Chap- 


ter XIII). Does the experiment show that the motor takes energy 
from the current while running? 

7. If there is a shop in your city where electrical apparatus, such 
as motors, is built, or a store where such things are sold, visit it and 
find out .what you can. Many interesting electromagnetic devices, 
toy motors, etc., are sold even in small electrical supply stores, and 
tradesmen are usually willing to explain them to any one who is inter- 
ested. Make a short written report of what you learn. 

8. What can you find out about trolley car motors, and how their 
power is transmitted to the car wheels? 

9. The Central Scientific Co., Chicago, sell the parts of a small motor, 
ready to put together. If you can not make a motor entire you can 
easily assemble one of these. 

10. If you know of a new house, where wires for electric bells and 
annunciators are being put in, go in and find out how the wires are 

11. Find out, if you can, what changes should be made in the con- 
nections of a motor in order to make it turn in the opposite direction. 

12. Read A Century of Electricity by Prof. T. C. Mendenhall 
(Houghton, Mifflin & Co., Boston), a very attractively written book, 
with much about the history of discovery. 



9. Sonrce of Current. In the preceding chapter we have 
seen how the modem electric motor might have been developed 
from principles that had all been discovered as long ago as 1825. 
Why was it that sixty years elapsed before it really grew into a 
practical machine, and came into general use? The answer is, 
that it had to wait for its counterpart, the d3niamo electric ma- 
chine. And why? Because no matter how efficient the motor 
itself may be, it must get its energy from the electric current. 
The only means then known of supplying electric currents were 
the various forms of voltaic cells, all of which derive their energy 
from the chemical combination of zinc with oxygen, just as the 
steam engine gets its energy from the chemical combination of 
the carbon and hydrogen of the fuel with the oxygen in the air. 
But the cost of zinc is so much greater than that of coal, gas, or 
oil, that it costs a great deal more to do mechanical work with a 
motor that is run by burning zinc in a battery, than it does to ido 
it with an engine that is run by burning coal or gas under a boiler. 
Thus the motor is of little practical value unless we can generate 
electric currents in large quantity and at reasonable expense. 
Where shall we look for the solution of this problem? 

230. Current and Magnetic Field. In our studies thus far 
we have found that many physical processes are reversible. Thus 
a current of air will turn a windmill and do mechanical work. 
Conversely, if we do mechanical work in turning a windmill back- 
wards, we can make it act as a rotary fan, and produce a wind for 
ventilating purposes. Heat may be converted into mechanical 
work. Conversely, mechanical work may produce heat. There- 
fore the question naturally arises: Since a current generates a 
magnetic field, can not a magnetic fisld he made to generate a cur- 



231. Faraday's Discovery. The discovery of how a current 
can be generated with the help of a magnet was made by Michael 
Faraday (1791-1867) in 1831. We shall be able better to ap- 
preciate this great discovery if we repeat some of Faraday's ex- 
periments. In order to do this we shall need a coil S, of many 
turns of fine wire, a bar magnet M, a couple of voltaic cells or 
other source of steady current, a sensitive galvanometer G, and a 
pocket compass. This apparatus. Fig. 137, differs in no essential 
way fr^m that used by Faraday. 

Before making experiments, let us see if our previous study will 

Fio. 137. The Moving Magnet Generates a Current 

enable us to foretell what results we may expect. We know 
that if we pass the battery current through the coil S, one of its 
ends will become a north-seeking magnetic pole, and the other a 
south-seeking pole. Lines of force will emerge from the former 
and enter the latter. Let us send the current from the battery 
in such a direction through the coil S that its upper end repels 
the north-seeking pole of the compass needle, and is therefore, 
itself a north-seeking pole. 

Now insert the galvanometer G into the circuit, and note 
its deflection. Suppose it is toward the right. We then know 


that when the galvanometer is deflected to the right, the current 
circulates in such a direction in the coil that its upper end is a 
north-seeking pole. Therefore, when the current is passing 
in the coil, it is able to do the work of pulling a south-seeking pole 
into the coil. If the phenomenon is reversible, we may expect 
that if there is no current flowing in the coil, and if we do the 
mechanical work of pulling the south-seeking pole out of the coil, 
we shall generate in the coil a current, and that it will flow, in the 
same direction as the current from the battery flowed. « 

232. Current Induced by a Moving Magnet. In order to 
verify this conclusion, we must remove the battery from the cir- 
cuit, .leaving the coil and galvanometer connected as before. 
Thert place the magnet inside the coil with its south-seeking pole 
down, and see that the ga,lvanometer is at rest in the zero posi- 
tion. When the magnet is quickly pulled out, the galvanometer 
gives a quick right-handed deflection. Qur prediction was correct. 
While the magnet is moirfng, a current is pissing in the coil. Since 
the deflection is right-handed, we knew tliat the current which we 
induced in the coil must be in the same direction as the battery 
current. That current made the upper end of the coil a north- 
seeking pole. The induced current, therefore, does the ,same. 

But since the upper end was a north-seeking pole, it tended 
to pull the magnet in, i.e., to stop its motion; so the induced 
current did what it could to oppose the motion by which it was 
generated. This is exactly what we ought to expect; for have we 
not learned long ago that a perpetual motion machine is impossible, 
and that if we produce some energy, as we have just done in 
generating this current, we must do some extra work in order to 
produce it? The extra work that we do is that of overcoming 
the magnetic attraction between the induced current and the induc- 
ing magnet pole. 

Faraday was much puzzled, at first, by the fact that the in-' 
duced currents were but momentary, for in his time the principle 
that the energy stored and the work done in storing it are always 
exactly equivalent to each other, was not so well known. It 
ought to be perfectly plain to us, however, that the induced 


current can last just as long as the work continues, and no 

We have jusi learned that when a south-seeking pole is with- 
drawn from the coil, the upper end of the coil becomes a north- 
seeking pole, which thus opposes the motion. What will happen 
if we push the south-seeking pole of the magnet back into 
the coil? If a current is induced in the coil, and if the direction 
of this current is such as to oppose the motion, it should make 
the upper end of the coil a south-seeking pole. When we try 
the experiment, we find that the galvanometer gives a quick 
deflection — not to the right, but to the left. But since a deflection 
to the right means that the upper end of the coil is a north-seeking 
pole, this left-hand deflection tells us that this upper end was a 
south-seeking pole, as we predicted. 

If we reverse the mjignet a/nd push its north-seeking pole 
into the coil, we get a right-hand deflection, indicating an induced 
current which makes the upper end of the coil a north-seeking 
pole, and which thus repels the approaching north-seeking magnet 
pole. When we withdraw the north-seeking pole of the magnet 
from the coil, we get a left-hand deflectiony indicating an induced 
current, which makes the upper end of the coil a south-seeking 
pole, and thus tries to attract the north-Peking magnet pole and 
oppose the motion of withdrawing it.* 

Hence we conclude that when a magnet pole is pushed into a 
coil of wire, or withdrawn from it, a current is generated in thai 
coil. This current lasts only while the motion lasts, and is always 
in such a direction that its magnetic field opposes the motion, 

233. The Number of Lines of Force is Changed. A little reflec- 
tion will enable us to see clearly that by all the four motions which 
we made with the magnet, we either pushed lines of force into the 
coil or pulled them out. We may, therefore, often find it con- 
venient to conceive that the induced current is generated by 
changing the number of lines of force that pass in one or the other 
direction through the closed conducting circuit. 

W'e should always remember, however, that when we vary the 
number of lines of force in any of these ways, we must expect to 



expend some energy; for if we could vary them without energy, 
and thus induce a current, we should be able to design a suc- 
cessful perpetual motion electrical machine — a 'thing which all 
competent minds agree is impossible. 

234. Currents Induced by Currents. In the last chapter we 
learned that a coil through which a current was flowing had mag- 
netic poles, just like a magnet. We may, therefore, expect that 
if we move such a current-bearing coil either into or out of another 
coil, we shall get effects precisely similar to those obtained by 
moving the magnet. The apparatus for the experiment b shown 

Fig. 138. The Moving Con^ Mat Generate a Current 

in Fig. 138. When we pass the current through the coil P, and 
then bring it quickly near the coil S, the galvanometer gives 
deflections as before, and these deflections again indicate induced 
currents, which in all cases oppose the changes that we make in 
the lines of force. 

We have learned, then, that a current in one closed circuit 
can be made to induce a current in another closed circuit. The 
first current is often called the primary or inducing current, 
and the second the secondary or induced current. 



235. Iron Core. Since we know that more lines of force can 
be gathered into the primary coil P by placing soft iron therein, 
a new question is suggested, What effect will this stronger field 
have on the secondary circuit? Will it vary the directions or the 
magnitudes of the induced currents? The answer ought to be 
easily forthcoming, for we know that there will then be more 
lines of force to move into the coil or out of it, but that their direc- 
tions will be the same as before. Therefore it is fair to infer 
that a greater number of lines, moved in the same time, will induce 
a stronger current each time, but that its direction will be the 
same as that induced by the primary current without the soft iron 
core C Again the appeal to experiment confirms our predictions. 
The greater deflections of the galvanometer indicate the presence 
of greater secondary currents. 

236. Currents Induced by Making and Breaking Circuits. 
There remains still another thing to try. By changing more 

Fig. 139. Currents are Induced When the Circuit is Closed or Opened 

lines in the same time, we change the number of lines at a greater 
time rate, i.e., more quickly. Is there any way in which we may 
further increase this rate of change? Evidently we can da so by 

250 PHY&ICS 

moving the primary coil more quickly, and on trial we find that 
the induced currents are still greater. 

But we have not yet reached the limit of increasing the rapidity 
with which the number of lines is changed. We can change 
them from zero to maximum, or vice versa, in a very small frac- 
tion of a second by placing the primary inside the secondary, and 
simply making and breaking the primary circuit. We may 
verify this deduction as we did all the others (Fig. 139). 

237. The Laws of Induced Currents. We are now ready to 
formulate the results obtained by this interesting series of ex- 
periments. They are : 

1. Whenever the number of lines of force that pass in a 
given direction through a closed conducting circuit is changed, a 
current is indujced in that circuit. 

2. Other things being eqv^l, the magnitude of the induced 
current is directly proportional to the rate at which the number 
of lines of force is changed, 

3. The direction of the induced current is xd'Ways such that 
its magnetic force opposes the motion which produces it. 

The third law is known as Lenz's law, from the name of the 
Russian physicist, Heinrich Lenz, who first announced it. It 
may here be mentioned that when a current is induced in a sec- 
ondary coil by starting a current in the. primary, the reaction of 
the secondary current stops the primary, if it ican not push it 
away mechanically; and the current induced by stopping the 
primary tends to keep the primary going. The reaction, therefore, 
may be electrical as well as mechanical. 

We are now in possession of all the principles necessary for 
the invention of the dynamo, the induction coil, the alternating 
current transformer, and the telephone; therefore we shall now 
take up each of these inventions in turn. 

238. The Dynamo Principle. In the last chapter we learned 
about the construction and action of an electric motor, and saw 
how it may convert electrical energy into mechanical work. In 
the present chapter we have learned that this process is reversible, 



Fig. 140. The Dynamo Diagram 

and that we can convert mechanical energy into electrical 

But how are the laws of induced currents applied in the con- 
struction and operation of the dynamo? With the aid of Fig. 140, 
let us try to find out. In the diagram, N and S represent the 
two poles of the field 
magnet, their lines of 
force being indicated 
by the long vertical ar- 
rows. FFi represents 
a single coil armature, 
CC a split ring com- 
mutator, and B-{- and 
B— a, pair of brushes 
attached to the termi- 
nals of the external 
circuit around which the current is to be sent. The shaft and 
bearings (c/. Figs. 129 and 130) are omitted for the sake of clear- 

The armature coil being in the position shown, the greatest 
possible number of lines- of force pass into the upper face F. But 
when the armature is turned through a quarter of a revolution in 
the direction of the curved arrows (seen at the right of the dia- 
gram), its plane will be vertical. The number of lines of force pass- 
ing into the face F will have been reduced from the maximum to 
zero. Therefore an induced current will circulate around the coil 
in such a direction as to make this face F a south-seeking pole, 
which, by its attraction, opposes the rotation. 

This current, whose direction is shown by the short, horizontal 
arrows, will charge brush B+ positively and 5— negatively, and 
will flow onward from B+ around the external part of the circuit, 
returning to the armature through brush B— and commutator seg- 
ment C. When the armature has turned through the second 
quarter revolution, its plane will again be horizontal, and it will 
again embrace the maximum number of lines of force. But since 
the lines now enter the face F^, and since pushing the lines into 
the face F^ has the same effect as has withdrawing them from F, 



the resulting induced current will continue to flow around the 
coil in the same direction as before. Therefore the brush B+ will 

again be positively 

During the third 
and the fourth quar- 
ter revolution the 
lines are withdrawn 
from the face F^ and 
pushed into the face 
F. Therefore the 
induced current 

around the coil is 
reversed during these 
two quarter turns. 
But it does not re- 
verse at the brushes, 
for at the instant 
when the third quar- 
ter turn begins, the 
commutator segments reverse their contacts with the brushes, and so 
B+ continues to be charged positively, and B— negatively. There- 
fore the current flowing around the 
external circuit is in the same 
direction throughout the rotation, 
i.e., from B+ to B—. This is 
the principle of the direct current 

239, The Dynamo. The power 
and efficiency of a dynamo are 
increased by the means previously 
described in the case of the mo- 
tor. The field magnets are elec- 
tromagnets, and instead of two poles there may be four or more. 
They are designed so as to give as strong and dense a field as 
possible (Fig. 141). 

Fig. 141. Six-Pole Field 

Fig. 142. Armature and Com- 



The armature consists of many coils wound on a soft iroti 
core. Not only must the armature be carefully balanced me- 
chanically, but the distribution of the coils must be such that the 
moments of the magnetic forces are also symmetrically balanced 
about the axis; otherwise the rapidly rotating armature will wabble 
like an ill-balanced flywheel (cf. Art. 91). Furthermore, the 
coils must be wound in slots in the core, and strongly bound in 
their places; for if they were not held firmly in the slots, the mag- 
netic forces that tend to stop 
their motion would combine 
with the centrifugal force to 
pull them out of their places 
(cf. Art. 90). The insulation 
of the coils should also be as 
perfect as possible. The 
iron between the slots also 
serves to fill the air gaps be- 
tween the coils, and conduct 
the lines of force into the 
space where they are most 
elTcctive. Fig. 142 shows 
the armature, with the com- 
mutator on the left. ' In Fig. 
143 the assembled machine 
is shown with the commu- 
tator, brushes, shaft and one 
of the bearings on the 
right. Fig. 144 shows how an armature core for a very large 
dynamo is built up of thin slotted plates of soft iron. 

240. Winding of the Field Magnets. In the direct current 
dynamo the current generated by the armature is used to excite 
the field magnets. Accordingly these are called self-excited, 
to distinguish them from the alternating current machines, which 
must be separately-excited by a direct current from a small 
separate dynamo. In the direct current dynamo, as in the motor, 
the armature current may be carried around the field coils from 

Fig. 143. Complete Dynamo 



Fig. 144. Building an Armature Core 

the positive brush, then to the external circuit, and thence back 
to the negative brush (series winding, c/. Figs. 134 and 152); or 

it may divide at the 
brushes, one branch 
going around the field 
coils, and the other 
around the external 
circuit (shunt winding, 
Figs. .135 and 153); or 
both styles ' of winding 
may be used together 
(compound winding, 
Fig. 136). Fig. 141 
shows the long, shunt 
coils, next the iron-clad frame and the short, series coils close 
to the ends of the pole-pieces. It is therefore a compound wound 

241. How the Field of Force is Built up. As the field magnet 
cores are of soft iron, it may be asked. Since the cores are not 
magnetized unless the current is flowing around them, and since 
a current can not be induced in the armature unless lines of force 
from the field magnet pass through it, how i^ it that the current 
can start at all? The answer is, that the field cores always 
retain a little of their magnetism after having once been strongly 
excited. This residual magnetism is sufficient to generate a 
small induced current in the revolving armature, and this small 
current, in turn, increases the magnetic strength of the field mag- 
nets. They are then able to induce a still stronger current in 
the armature. The magnetism of the field cores and the resulting 
current in the armature thus add gradually each to the strength 
of the other, until the field magnets are saturated. 

242. Magnetos. The early dynamos were magnetos, i.e., 
their field magnets were permanent steel magnets, resembling 
that in Fig. 140. Such machines are still much used in operating 
call bells on private telephone lines, and in producing sparks for 
the ignition of the gases in gas engines. 



243. Alternating Gnrrent Dynamos. For many purposes, 
an alternating current has very decided advantages over a direct 
current. In general principle the ''alternator" resembles the 
direct current machine, but it has collecting rings (c/. Fig. 
128) instead of comniutator segments, so that the electric impulses 
sent out to the line change direction every time a pair of its 
coils passes a pair of its poles. As there are usually several pairs 
of poles and as many pairs of coils, there will be several alterna- 
tions at each revolution. 

244. The Induction Coil. The induction coil (Fig. 145) 
is an instrument frequently mentioned in the papers and maga- 

FiG. 145. The Induction Coil 

zines, because it is used in producing X-rays, and in starting the 
ether waves by which messages are sent through space without 
wires. It was invented by an American, Charles G. Page, in 
1838. It consists of a primary coil P, of a few ttirns of coarse 
wire, containing a soft iron core, and surrounded by a secondary 
coil Sy of many turns of fine wire, whose ends lead to a pair of 
insulated knobs or points TT, Thus far it is precisely like our 
apparatus for investigating induced currents (Fig. 139). 



But for convenience and speed in making and breaking the 
primary circuit, there is usually added an automatic contact 
breaker Hy which keeps itself vibrating, and automatically opens 
and closes the primary circuit exactly as the armature of the 
electric call bell does (Art. 218), Alternating currents are thus 
induced in the secondary coil. 

Sincea current impulse or pressure, called electromotive force, 
is started in every turn of the secondary coil, every time that the 
primary circuit is made or broken, it follows that these impulses 
in all the turns will be added together. Therefore, up to a certain 
limit, the pressure of the induced current increases with the num- 
ber of turns in the secondary coil. The induced electromotive 
force is also proportional to the suddenness with which the pri- 
mary current is started or stopped (c/. Art. 236). As the alter- 
nating induced currents surge back and forth in the' secondary 
coil, the electrical pressures at the terminals TT become so 
great that disruptive discharges occur between them. A 40-inch 
spark coil produces a pressure equal to that of from 00,000 to 
iOO,000 voltaic cells and contains over 250 miles of wire in the 
secondary coil. Large induction coils must be designed with 
great care, especially with regard to the insulation, which would 
otherwise be punctured by the great electrical pressures. 

245. The Alternating Current Transformer. Fig. 146 rep- 
resents the apparatus used by Faraday in one of his earliest ex- 
periments with induced 
currents. It will be 
seen that when a cur- 
rent is started in one of 
\he coils P, it will send 
its lines of force around 
through the iron of the 
ring C, and thut when 
these lines of force enter 
the other coil S, they will induce in it a current whose direction 
is opposite to that of the primary and which may be detected 
by the galvanometer G. If the primary current stops, a cur- 

FiG. 146. Faraday's Ring 



Fig. 147. A Trans- 

rent is induced havlpg the same direction as the primary. If 

the primary current is alternating instead of intermittent, its 

lines of force will enter the secondary coil first 

from one direction and then from the other 

alternately, and will thus produce alternating 

currents in the secondary. . Since the elec- 
tromotive force or pressure in the secondary 

coil is proportional to the number of turns 

of wire in it, it follows that the electromotive 

forces in the primary and secondary coils are 

proportional to the corresponding numbers 

of turns of wire in the two coils. For ex- 
ample, if the primary coil has 100 turns and 

the secondary 100,000, the electromotive force 

of the induced current will be 1000 times as 

great as that of the inducing current. 

Thus we may send an alternating current 

of low pressuie and large quantity into the 

short coil of such a "transformer,'* and get 

out of the long coil an alternating induced current at high 

pressure, and of proportionally smaller quantity. Conversely, we 

can send an alternating current of high 

* M pressure into the long coil, and get out of 

"' the short coil an alternating current of 

lower pressure and proportionally larger 
quantity. If used in the former way, 
the apparatus is called a "step-up" 
transformer; if in the latter way, a 
"step-down" transformer. The induc- 
tion coil is a "step-up" transfonner. 
^^^^^^^^ Transformers are used extensively in 

P?|8^^^^^B electric lighting and car service, because 
"^^^H the current can be transmitted with far 

greater economy at high pressure than 
at low pressure. High pressure cur- 
rents, however, arc not suitable for use in lamps, and are not 
permissible in buildings, because of the danger of fire and loss of 

Fig. 148. CoNSTRurTioN OP 
THE Transformer 


life. High pressure currents from alternating dynamos are, 
therefore, distributed to transformers similar in construction to 
Faraday's ring. These are placed on poles outside the buildings, 
Fig. 147, and the low pressure currents are carried into the 
buildings, for service either in electric lamps or in alternating 
current motors. Fig. 148 shows such a transformer with the 
outside case taken off. 

246, Alternating Current Motors. There are two kinds of 
altej-nating current motors. Synchronous motors resemble 
alternating current dynamos in construction. They are so called 
because the alternating currents in their fields and armatures 
keep time, or step, with those in the dynamo that funiishes the 
current to them. The armatures of induction motors are turned 
by the magnetic forces acting between the field currents and the 
resulting induced currents in the armature. They require very 
little care, because the armature currents* have no electrical con- 
nection with the supply wires, and therefore they have no sliding 
electric contacts. 

The study of alternating currents and induction motors, 
though exceedingly interesting, is beyond the scope of an ele- 
mentary course. 

247. The Telephone. How is it that sounds so complex 
as those which are produced by spoken words, with all their vari- 
ations of loudness, pitch, and quality of tone, can be taken up 
by a small piece of sheet iron and transformed into electrical 
waves? And how is it that these electrical waves, after traveling 
along hundreds of miles of wire, can be retransformed into sound 
that is a close copy of that produced by the voice of the speaker? 
This is, indeed, the most marvelous of all the facts that our studies 
in Physics have yet brought to our attention. And the more we 
learn of sound, and the great complexity of the motions which it 
impresses on the air, the more we shall be led to wonder, that a 
pair of such simple contrivances as a telephone transmitter and 
receiver can work such a miracle. 

The construction of the telephone is easily understood. The 
parts are shown in Fig. 149. First, there is a funnel-shaped 



mouthpiece M, iivto which we talk. This mouthpiece keeps 
the sound from spreading into space, and directs it against a 
diaphragm or disc of sheet iron Z), placed just at the end of the 
funnel. Behind the diaphragm D, and attached to it, is a smaller 
disc of carbon E, and just behind £ is a second disc of carbon 
E, which is attached at its back to a metallic plate B, The small 
space between E and E is filled with grains of hard carbon. 

The metal supports on which the carbon discs E are mounted 
are insulated from each other; but one is electrically connected 
with one of the terminals of a voltaic battery Ba, or other source 
of current, while the other is connected with one of the terminals 
of the primary wire of a small step-up induction coil I. The other 

Fig. 149. Telephone Transmitter, Receiver, and Circuit 

terminal of this primary wire is connected with the other ter- 
minal of the battery. A current of electricity is thus always pass- 
ing between the carbon discs £, across the loose contacts of the 
carbon gi'anules, and completing its circuit back to the battery 
by the way of the primary wire of the induction coil. 

Now, a loose contact between two pieces of conducting matter 
has the very remarkable property of conducting a current better 
when the pressure on it increases. Very small variations in pres- 
sure at the points of contact produce changes of considerable 
magnitude, in the strength of a current of electricity that may be 
passing across the contact. Such a loose electrical contact is called 
a microphone contact because it enables us to hear very faint 


sounds. The two carbon discs with their microphone contact, the 
'battery, and the induction coil, are the only essential parts of the 
telephone transmitter; but the metallic case is of course necessary 
to protect these parts and hold them in their places. 

The receiving instrument, or Bell telephone, is even more 
simple. Its only essential parts are a strong steel magnet if, 
whose poles are surrounded by coils of fine wire CC, and a dia- 
phragm D' of sheet iron like that of the transmitter. Even 
this diaphragm is not absolutely necessary, for the sounds 
can be heard without it. The instruments are connected, as 
shown, the diagram for the other station being exactly like Fig. 
149, except that it is reversed. Their action is partially explained 
as follows: 

1. The sound sets the diaphragm D of the transmitter 
into vibration. 2. In some way not yet completely under- 
stood, these vibrations, when communicated to the micro- 
phone contacts, produce variations in the electric current flowing 
through them. These variations are counterparts in every detail 
of the sound vibrations. 3. These variations of current strength 
produce corresponding variations in the number of lines of force 
passing through the secondary of the induction coil. 4. The 
result is that induced electrical surges are sent chasing one another 
along the line ^ire to the distant receiving instrument. 5. These 
electrical waves, surging backwards and forwards in the little 
coils of the receiver, produce corresponding variations in their mag- 
netic field. These variations in the number and direction of 
the lines of force that pass through the coils cause mechanical 
vibrations of the diaphragm D', 6. Finally, the vibrations of the 
diaphragm are transmitted to the air, reproducing all the modula- 
tions of tone quality, pitch, and loudness that belong to the 
sounds emitted by the speaker. 


1. The electrical energy of a battery is obtained by consuming 
zinc, and the voltaic cell is therefore too expensive for generat- 
ing electricity on a large scale. 

2. Many physical processes are reversible, and we find that 


this is true of the conversion of electrical energy into mechanical 

3. Whenever the number of lines of force that pass through a 
conducting circuit is changed, an induced current is started, 
which lasts only while the change is going on. The magnitude 
of the induced electromotive force is proportional to the rate of 
change in this number of lines of force, and its direction is always 
such that its reaction — either mechanical or electrical — opposes 
the change that causes it. 

4. The dynamo electric machine is a contrivance for the 
conversion of mechanical energy into electrical energy. It con- 

.sists of a field magnet and an armature, one or the other of which 
rotates, and a sliding contact device. 

5. In the direct current dynamo the sliding contact device is 
a commutator and brushes; in the alternator it is a pair of col- 
lecting rings and brushes. 

6. Each coil of the annature, as it rotates, generates alter- 
nating induced currents, which are sent out through collecting 
rings, or a pair of commutator segments, and thence by a pair 
of brushes to the external circuit wherein the electrical work is to 
be done. 

7. The field magnets of a direct current dynamo are excited 
by the current from its own armature, and they may be series 
wound, shunt wound, or compound wound. 

8. The field magnets of an alternator must be excited by a 
separate direct current dynamo. 

9. An induction coil consists of a primary coil which contains 
a soft iron core and is surrounded by a secondary coil. The 
primary current is made and broken by a contact breaker. Such 
a coil may be made to give long, powerful sparks. 

10. Alternating current "step-up" transformers are used to 
transform currents of low pressure and great quantity into cur- 
rents of higher pressure and smaller quantity for transmission 
to a distance. "Step-down** transformers are used to reconvert 
these into currents of low pressure and large quantity for use in 
lamps and motors. 

11. A telephone transmitter consists essentially of a diaphragm, 


a microphone contact, and a small induction coil, in circuit with 
a local battery or other source of direct current. 

12. A telephone receiver consists of a steel magnet, a small 
coil (or two coils) of wire, and a diaphragm. 

13. Sound impulses are transformed by the transmitter into 
electrical waves, which are sent along a wire to a distant receiver. 

14. The receiver reconverts these electrical waves into soiind 
impulses similar in every detail to that which was spoken against 
the diaphragm of the transmitter. 


1. Mention some physical processes that are reversible. 

2. Describe the four ways in which currents may be induced in a 
coil of wire by means oT a steel magnet. 

3. In each of these cases, what is the kind of f>ole induced at the 
end next the magnet, the direction of the force between the induced 
and hiducing poles (i.e., attraction or repulsion), and finally, the effect 
on the motion (i.e., assistance or opposition)? 

4. In a similar manner, describe the eight different ways in which 
an induced current can be started in a secondary coil by means of a 
primary coil and a battery. 

5. What variation in these effects will result from the use of a soft 
iron core? From increasing the suddenness of the motions? 

6. Explain why the electromotive forces of induced currents are of 
short duration. 

7. State the lav/r, of induced currents, in which the results of all such 
experiments arc summed up. 

8. What are the essential parts of a dynamo-electric machine? 
Point out the resemblance in construction, and the difference in action, 
between a direct current dynamo and a direct current motor. 

9. Briefly explain how the act of rotating an armature coil sets up 
therein an induced current which changes direction at every half turn. 

10. Explain ho'^V these alternating currents may be led out to the 
external circuit as alternating currents by means of collecting rings, 
or converted into direct currents by means of a commutator. 

11. How are the magnitude and the uniformity of such direct 
currents affected by having the armature made up of several coils 
instead of one? 

12. Mention advantages gained in the design of a dynamo by the 
following features: (a) increasing the number of field poles; (6) per- 
fectly balancing the coils electrically and mechanically; (c) slotted 
armature core; (d) perfect insulation; (e) good ventilation. 


13. Describe the three modes of field magnet windings. 

14. Explain how a self-excited machine builds up its own mag- 
netic field. 

15. What are magnetos, and what are some of their uses? 

16. Diagram an induction coil, trace the primary current through 
its circuit, and show how the induced currents of the secondary coil 
are started. Explain the action of an automatic break hammer. 

17. Describe the construction and operation of the alternating 
current transformer. State the relation of the electrical pressures to 
the numbers of turns in the two coils. What is the great advantage of 
such transformers in alternating current lighting and power circuits? 

18. Diagram a telephone circuit with a transmitter and receiver 
at each end of the line. 

19. Trace the local current in the transmitter, and describe the 
manner in which its strength is varied in correspondence with the 
variations of air pressure due to the sound. Describe the induced 
currents that result from these variations of the local or primary current. 

20. Describe the results of these induced currents when they reach 
the receiving telephone. 


1. Examine a "spark coil," or "kicking coil," such as is in common 
use for lighting gas by electricity. It has a soft iron core and only one 
coil, of many turns. When placed in a circuit with several battery 
cells, it gives a strong spark on breaking the circuit, whereas if the wire 
were not coiled no spark could be obtained. Does this imply' that an 
induced current is generated "at break" which adds its electrical 
pressure to that of the battery? May a current in each turn of this coil 
induce a current in every other turn? On "making" the circuit, would 
the induced current be in the same direction as the battery current, or 
in the opposite direction? This added current is called a self-induced 
current. Does it differ essentially from any other induced current? 

2. A break hammer induction coil is subject to troublesome spark- 
ing at the break. Is this spark due to the same cause as that of the 
kicking coil? This spark, by forming an arc, like that of an arc light, 
bridges the gap, and not only burns the contacts, but also prolongs the 
time of breaking the primary circuit. Will the induced current in 
the secondary be as strong as if the arc were not formed? The arc may 
be partially prevented by connecting the opposite coatings of a con- 
denser across the gap. Why? 

3. When two telephone wires, whose circuits are completed through 
the ground instead of by return wires, are placed on poles parallel to 
one another, the conversation on one line may be heard on the other. 
Can you explain why? The Jipise of the trolley cars and of telegraph 


instruments is often heard in the telephone. May this be due to the 
same cause? 

4. Look carefully at Plate III. Do you see a dynamo direct- 
connected to it? Find the iron-clad field magnet, and the armature 
Also examine Plate VII. This is an alternating current generator. 
Find the armature and the field poles. In this dynamo, is it the field 
or the armature that revolves? 


1. If you have made yourself a galvanometer, as was suggested in 
Chapter XI, repeat the experiment of Faraday's ring (Art. 243); and 
also one made by Henry at about the same time (c/. Hopkins's Ex- 
perimental Science, pp. 467-476). 

2. Visit the power house of the electric lighting company or the 
street railway company, and make a brief report on the results of your 
investigation. (In most cases it will be necessary to write a letter to 
the manager, stating why you wish admittance and requesting the 
favor of a pass.) 

3. Belt a toy motor to the flywheel of a sewing machine, so that 
you can rapidly turn the armature by means of the treadle. Connect 
the terminals of the motor with your galvanometer and work the 
treadle. Does the galvanometer indicate that the motor is operating 
as a dynamo? If you have not made a motor you can buy one at an 
electrical supply store for a dollar or less. 

4. Unscrew the diaphragm end of a receiving telephone case, re- 
move the diaphragm and look at the end of the magnet and its coil. 

5. Read Tyndall's Faraday as a Discoverer (Appleton, N. Y.), and 
S. P. Thompson's Life of Faraday (Macmillan, N. Y.). 

6. Look up the biography of Joseph Henry. See, "A Study of the 
Work of Faraday and Henry,*' by Mary A. Henry {Electrical Engineer, 
N. Y., Vol. 13, p. 28), also Cajori's History of Physics. . This last book 
will give you references to books on the lives of all the discoverers in 

7. Forbes 's Elementary Lectures on Electricity and Magnetism (Long- 
mans, N. Y.), and Wright's The Induction Coil in Practical Work (Mac- 
millan, N. Y.), are especially interesting in their descriptions of the 
phenomena of induced currents. 

8. If Faraday's Experimental Researches in Electricity is in your 
city library, read some of it. It is one of the most remarkable books 
that was ever written. It will be worth your while to become at 
least a little acquainted with the mind of this great man. 



248. ftuestions for Further Study. In the preceding chap- 
ters, we learned how a dynamo and a motor work; but some ques- 
tions of great interest still remain unanswered. How do arc 
lamps and glow lamps work? How much power is required to 
operate them? What are the methods of distributing the cur- 
rent? How do electricians measure currents and calculate their 
power? How much loss is caused by the resistance of the wires? 
What uses can be made of the heat developed by the current? 
How is electroplating done? How does a storage battery differ 
from a voltaic battery? These are things that everybody wants 
to know something about, and a little further study will enable 
us to understand them. 

249. The Arc Light. In 1808, Sir Humphry Davy, the 
predecessor of Faraday at the Royal Institution, produced the 
first arc light with a powerful battery and two pencils of carbon. 
When these two carbons, connected with the terminals of the 
battery, were brought into contact and then slightly separated, 
the current was not broken, because an arc, composed of white 
hot vapor of carbon, was formed. The carbons, being in contact 
with the air, are gradually burned up, just as is the carbon in 
burning illuminating gas or oil. In all arc lamps an ingenious 
arrangement of electromagnets "feeds" the carbons together auto- 
matically as fast as they bum away (c/. Art. 270). 

250. Pressure and Current in the Arc Lamp. An ordinary 
street lamp is equivalent nominally to 2000 candles. It is found 
not to bum satisfactorily unless it is fed by a current having a 
constant strength of about 9.5 amperes, the electrical pressure 



at its terminals being maintained at about 50 volts. Lamps of 
greater candlepower must, of course, have more current. We 
have already learned something in a general way about cur- 
rent strength, resistance, and pressure; but if we wish to know 
anything of the way in which electrical calculations are made, 
we must learn how to express our ideas more precisely. Some 
definitions and precise statements of relations, therefore, are 

251. The Current Strength in a conductor is the rate of flow 
of the current, i.e., the quantiiy of electricity passing per second 
at a given cross-section of the conductor; and the ampere, the 
practical unit of current strength, is defined as the steady current 
which deposits silver by electrolysis from a solution of a silver 
salt at the rate of .001118 grams per second (c/. Art. 208). 
The ampere is named in honor of Andr6 Marie Ampere (1775- 
1836),. who was professor in the Polytechnic School at Paris, and 
who followed up the discovery of Oersted with valuable researches 
on relations between currents and magnets. 

252. Eesistance. We are accustomed to conceive that a 
conductor offers resistance to the passage of a current of elec- 
tricity, and that an electromotive force, or electrical pres- 
sure, is required to force the current through it, because in trans- 
mitting the current, the conductor absorbs some of the electrical 
energy and gives off this energy again in the form of heat. The 
unit of resistance is the ohm, which is defined as the resistance at 
0° C. of a column of pure mercury 106.3 centimeters long and 
of a uniform cross-sectional area of 1 square millimeter. Such a 
column should weigh 14.45 grams. The ohm is named in honor of 
Georg Simon Ohm (1789-1854), an eminent German physicist, who 
was teacher of mathematics and physics at the Gymnasium at 
Cologne, and afterwards professor at the University of Munich. 
Ohm investigated, both experimentally and mathematically, the 
resistances of conductors and their relations to current strength. 
By his experiments and those of others, the following relations 
have been established: 


253. The Laws of Eesistance. The resistance of a conductor 

1. Directly proportional to its length. 

2. Inversely proportional to its cross-sectional area. 

3. Directly proportional to a constant whose value depends on 
the material of the conductor and on the units in which its length 
and cross-section are expressed. This constant is called the 
RESISTIVITY of the substance, and it represents the resistance at 
0° C. of a conductor of the given substance having unit length 
and unit cross-section. 

4. Other things being equal, the resistance of a given conductor 
depends on the temperature. For most metallic conductors, the 
resistance diminishes as the temperature is lowered; and it is 
interesting to note that according to some recent experiments, 
this diminution appears to take place at such a rate that at the 
absolute zero they would have no resistance (c/. Art. 123). For 
carbon, and for those substances that are broken up or electrolyzed 
when conducting a current, the resistance is diminished by raising 
the temperature. 

264. Ohm's Law. One of the most important contributions of 
Ohm to our knowledge of electric currents is the law known by 
his name, and stated as follows: The current strength in any 
conducting circuit is directly proportional to the electric 
pressure or electromotive force, and inversely proportional to the 
corresponding resistance. Letting C represent the current strength, 
E the electric pressure, or electromotive force, and R the 


resistance, we may express Ohm's law by the equation C = -^ or 

^ .. Pressure in volts ,^^^ 

Current m amperes = ^ — ;— -. ^ . (11) 

^ Kesistance m ohms ^ ^ 

This equation defines the volt for us ; for if the current strength 
and resistance are each made equal to 1 unit, the pressure, ac- 
cording to the equation, is 1 volt. 

A VOLT, therefore, is that electrical pressure or that electro- 
motive force which will maintain a current of one ampere in a 
conductor whose resistance is one ohm. The volt is named in 



honor of Volta (c/. Art. 207). A simple voltaic cell has an 

electromotive force of nearly one volt. A difference in 

electrical pressure is often spoken of as a difference of 


255. Ammeters and Voltmeters. It will interest us to learn 
how the current strength and pressure required by a lamp or 
a motor are measured. The instruments most widely used for 
this purpose are made on the principle of the D'Arsonval gal- 
vanometer. The coil is pivoted in jeweled bearings, and bal- 
anced against a hairspring after the manner of the balance wheel 

of a watch. In all voltmeters 
and ammeters, the number 
of volts or amperes is indi- 
cated by a pointer attached 
iT, to the coil and moving over 
a scale which has been ac- 
curately graduated or "cal- 
ibrated " by reference to currents of known 
strength, so as to read volts or amperes. The 
voltmeter has a very high resistance, and the 
ammeter a very low resistance; the voltmeter 
is connected as a switch or "shunt" across the 
terminals of the lamp or motor, where the 
pressure is to be measured; the ammeter is 
placed "in series" in any part of the circuit 
around which the current is flowing. Fig. 150 shows the proper 
method of connecting them. Fig. 151 shows a portion of the 
switchboard in a power-house, with ammeters, voltmeters, and 
switches. • 

Fig. 150. Voltmeter 
AND Ammeter Con- 

256. To Calculate the Power. The next problem that claims 
our interest is the calculation of the power used in the lamp. In 
Chapter IX we learned that we could calculate the amount of work 
done per second in the cylinder of a steam engine, not only by 
taking the product of the average force of the steam and the distance 
traversed per second by the piston, but also, more conveniently, 



by taking the product of the average steam pressure and the 

quantity (volume) of steam used per second. It is easy* to show 

that similarly the work done per second by an electric current is 

proportional to the product of the electrical pressure and the 

quantity of electricity 

used per second. If we 

measure the pressure in 

volts and the quantity- 

per-second, or current 

strength, in amperes, it 

is obvious that the power 

will be one unit when the 

pressure is one volt and 

the current strength one 

ampere; therefore, the 

following definition is 

adopted for the unit of 


The unit of electrical 
power or activity is the 
power of a current of one 
ampere under a pressure 
of one volt. This unit is 
called the watt, in honor 
of James Watt, the in- 
ventor to whom we are most indebted for the modem steam engine. 
A watt is found to be equal to 10^ ergs per second, or t^^^ horse- 
power (c/. Art. 43). With these units, the equation that expresses 
the power of a current is: 

Power in watts = Current in amperes X Pressure in volts (12) 

Or, if A represent the number of watts, C the current strength, 
and E the pressure, A = CE, 

With this equation we can now calculate the power consumed 
in our 2000 candlepower arc lamp, for since C = 9.5 amperes 
and E = 50 volts, the power .4 = 9.5 X 50 = 475 watts. Let the 
student find the number of ergs per second and the horse-power 
that correspond to this number of watts. 

Fio. 161. Switchboard 



257. Watt Meters. When large quantities of electrical energy 
are used in varying amounts, it is most convenient to measure the 
total amount of it in watt-hours. 

The watt-hour is the amount of energy furnished in one h6ur 
at the rate of one watt; and it therefore equals 10^ — X 3600 
sec = 36 X 10® ergs; 1000 watts is called a kilowatt (Y'^K. W.). 
The number of watt-hours Used by a consumer is measured by 
an interesting instrument called a watt meter. A common form 
of watt meter is a little motor having no soft iron cores. Its 
field coils have few turns, and are placed in series in the circuit 
whose energy is to be measured; the armature coils have many 
turns, and are connected as a shunt across the mains or feeders, 
like a voltmeter. The instrument is ingeniously regulated, so 
that the number of revolutions of the armature is proportional 
to the energy supplied. Therefore, if a train of clock wheels is 
geared to the armature, the total number of watt-hours of energy " 
that have passed the meter up to any given time may be indicated 
on a series of dials by index hands attached to the gear wheels, 
just as the number of cubic feet is indicated on the dials of a gas 





^ ♦. 


> " 


.«. ( 





'■ t 


258. An Arc Light Plant. We now have at command the 
knowledge that is necessary in making the calculations for a small 

arc lighting plant. Suppose that 
we wish to light a shop with ten 
2000 C. P. (candlepower) arc 
lamps, and must know the neces- 
sary engine and dynamo power. 
How shall we attack the problem? 
In a case like this the lamps 
are ordinarily placed in series as 
represented in the diagram. Fig. 
152; therefore the total resistance is the sum of all the resistances 
in the circuit. 

The loss in pressure, or fall of potential in any part of the cir- 
cuit is proportional to the corresponding resistance, m accordance 
with Ohm's law, i.e., E = CR. (cf. Art. 254). Therefore, the 


Fig. 152. Lamps in Series 


NUMBER OF VOLTS USED IN THE LAMPS is equal to that for one 
lamp multiplied by the number of lamps. So the pressure needed 
for the 10 lamps is 50 X 10 = 500 volts. 

We must now find the voltage necessary to overcome the 
resistance of the line. Since this voltage is not available for use 
in the lamps, it is called the line loss or "drop." Let us suppose 
that the total length of wire from the dynamo through all the 
lamps and back again to the dynamo is 600 ft. The fire insurance 
regulations require us to use at least a Number 14 wire to carry 
9.5 amperes (Table I, page 298). This, by reference to Table I, is 
found to have a resistance of 2.565 ohms per thousand feet. Since 
the resistance is proportional to the length, that of 600 ft. is 0.600 
of 2.565, or 1.5 ohms, nearly. The loss of pressure in the line i^, 
therefore, 9.5 amperes X 1.5 ohms = 14.25 volts. 

We have found that there are 500 volts required for the lamps, 
and that the line loss or "drop" is 14.25 volts; hence the electro- 
motive force demanded from the dynamo is 514.25 volts, and at 
this pressure it must send out 9.5 amperes. The output of 
power by the dynamo is 514.25 volts X 9.5 amperes = 4885 
watts. Since more lamps may be needed, as the require- 
ments of the shop increase, it is customary to provide for this 
increase of power. Let us suppose, therefore, that we are 
to order a dynamo capable of giving 750 volts and 9.5 amperes. 
The power of this machine will. be 7125 watts or, say, 7.5 K. W. 
The equivalent of 7.5 kilowatts in mechanical horse-power is 
VjV = 10 H. P., nearly. 

Since the efficiency of a good dynamo is about 90 per cent, 
we must allow for a 10 per cent loss of energy in the dynamo itself. 
There is also a further loss of about 5 per cent in transmitting the 
mechanical power from the engine to the dynamo. Hence, 10 
H. P. is 85 per cent of the power that must be furnished by the 
engine to provide for the greatest load that it will get from the 
electric plant. This extra power to be provided by the engine, 
then, is V/ of 10 = 11.8 H. P. Our engine must, therefore, be 
big enough to take care of a load of about 12 H. P. in addition to 
the power furnished by it for running the machinery of the 



259. Incandescent Lamps. The incandescent or glow lamp 
consists of a slender thread or filament of specially pre- 
pared carbon, enclosed in a glass bulb and mounted on a pair 
of terminals that pass through a glass plug at the bottom of the 

The air is pumped from the bulb, which is then fastened into 
a base. This base may be screwed into a socket in such a way 
that when the terminals of the socket are connected with the 
supply wires, the current is conducted through the filament. The 
carbon has a relatively high resistance, and when a sufficient 
current passes through it, the resulting heat makes it white hot 
or incandescent (c/. Art. 153). It can not bum, however, because 
the air has been removed from the bulb, and no oxygen is there 
for it to combine with. 

A glow lamp is usujdly so adjusted that it gives 16 candle- 
power when a current of 0.5 amperes is passing through it. Its 

resistance is 220 ohms; hence 
the fall of pressure through it 
is 110 volts, and it takes en- 
ergy at the rate of 55 watts. 
The student may easily verify 
the last two statements by cal- 
culation from equations (11) 
and (12), Arts. 254, 256. 

260. The Parallel Method 
of Distribution. The diagram , 
Fig. 153, shows the method generally employed for distributing 
the current to glow lamps. 

The dynamo is designed and operated so as to maintain a 
constant difference of potential at its terminals, whether the cur- 
rent taken from it is large or small; and in good practice the wires 
for the mains are so chosen that the pressure lost in traversing 
them shall not be more than about 10 per cent of that furnished 
by the dynamo. 

Since the loss in pressure through the lamps is 110 volts, the 
dynamo must at least have an electromotive force equal to VV 

Fig. 153. Lamps in Parallel. 


of 110 = 122.2 volts. For small plants the usual machine is wound 
for 125 volts. 

261. An Incandescent Light Plant. Let us suppose that a 
certain man has on his country place a good-sized waterfall, and 
wishes us to tell him whether he can use this power for lighting 
his house. How shall we apply our knowledge of pKysics so as to. 
solve his problem for him? 

We must ascertain: (1) The greatest number of lamps that 
will be in use at any one time. (2) The distance from the fall 
to the center of distribution, or the point where the branches to 
the various rooms are to be taken from the main wires or *' feed- 
ers." (3) The height of the falls. (4) The least volume of 
water per second that will be available for use. This makes it 
necessary to measure the cross-section and velocity of the stream 
above the falls. 

Let us suppose that he wishes to light 200 lamps, 150 of which 
are likely to be in use at any one time, and that the falls are lo- 
cated 1500 feet from the center of distribution. 

The lamps will require a pressure of 110 volts, and if a 125 
volt dynamo is used, the line loss allowable will be 15 volts, or 
12 per cent. Since the main current divides amongst the lamps 
it must equal the sum of the currents in all the lamps. There- 
fore the number of amperes required for 150 lamps of 16 C. P. 
each, is 0.5 X 150 = 75. The output, therefore, must be 125 
volts X 75 amperes = 9375 watts. In designing a lighting plant it 
is always well to allow for a few more lamps, so we will figure 
on a 10 K. W. dynamo, giving an output of 10,000 watts, or, in 
mechanical units, -^H^ = ^^-^ horse-power. 

Allowing an efficiency of 90 per cent for the dynamo, the 
mechanical H. P. supplied to it by the water wheel must be: 

W of 13.4 = 14.89 H. P., or, say, 15 H. P. 

The efficiency of a good turbine water-wheel is about 80 per 
cent, and if we allow for an additional loss of about 5 per cent in 
transmitting the power from the turbine to the dynamo, the power 
furnished by the water must be VV" of 15, or 20 H. P. 


Now let us suppose that the falls are 300 cm high and that 
by floating ^ stick on the water just above the falls and timing 
it with a watch, we find that it passes over a measured distance 
of 200 cm in 2 sec. . 

The velocity is, therefore, 100 ^. If in measuring the depth 
and width of the stream at the place where it goes over the falls, 
■ we find that a fair average of each dimension gives us, depth 60 
cm, width 300 cm, the average cross-sectional area of the stream is : 
18,000 cm^, and at the speed of 100 ^ the volume passing the 
falls in 1 sec is 18 X 10^ cm^. The mass of the water 
(cf. Art. 32) is 18 X 10^ gm, .and its weight is therefore 
18 X 10^ X 980 = 1764 X 10* dynes. Since the distance 
through which this weight can act is 300 cm, the potential 
energy of the fall is 1764 X 10* X 300 = 5292 X 10* ergs each 
second. Therefore, since 1 H. P. = 746 X 10^ ^, its horse- 
power is: ^f/^^l? = 72 H. P., nearly. 

We see that the inaccuracy in our method of measuring the 
falls is not serious, for the result of the calculation shows us that 
we have a large margin. We may, therefore, safely assume that 
the stream will furnish us the necessary 20 H. P. 

The sizes of wire for the feeders and branches still remain to 
be calculated. We have seen that our allowable line loss is 15 
volts. We must not have more than 3 per cent drop in voltage 
in our distributing wires, because the drop varies with the current 
used, so that when only a few lamps are lighted, it will be less than 
when all are lighted. Unless the drop allowed for were small, 
there would then be too much pressure for these lamps and their 
life would thus be shortened. We shall therefore take 3 volts 
for the drop in the distributing wires and leave 12 volts for drop 
in the feeders. Now, since the line drop is to be 12 volts and 
the current required is 75 amperes, we have from Ohm's law: 

R = jz = -^ : — = 0.16 ohms. Since the feeders are 1500 

C 75 amperes 

feet long, the length of wire in them is 1500 X 2 = 3000 feet; and 

the number of ohms per thousand feet is ^ J ^ = 0.0533. Consulting 

the wiring tables, we find that a Number 0000, Brown and 


Sharpens gauge wire has a resistance per 1000 ft. of 0.04966 ohms, 
which is the nearest to 0.0533 ohms and is, therefore, the size to 
be chosen. 

The sizes of wire on each of the various branches are chosen 
by a similar calculation, in accordance with the number of am- 
peres taken and the length of the branch wires, so as to give a 
drop in each group of nearly 3 volts, as demanded by the con- 
ditions stated. 

Such a lighting plant as this would be rather expensive; but 
the energy would cost nothing; the repair bills would not be large, 
no high-priced attendance would be necessary, and the only im- 
portant cost would be the interest on the money invested. 
The power could be utilized in the daytime, by means of motors, 
for operating a threshing machine, feed chopper, cream sepa- 
rator and churns, elevators, sewing machines, electric fans, and, in 
fact, for everything in which power is needed on the place, in- 
cluding the charging of electric automobiles. A few storage 
battery cells could also be kept charged by the current and used 
for operating door bells, burglar alarms, signal bells, and other 
household apparatus, and an electromagnetic device for stopping 
and starting the water-wheel by means of a switch, located in 
the house. 

Heating coils also might be used in the house for cooking, 
ironing and the like. Thus, considering the great convenience, 
cleanliness, and wide range of usefulness afforded by such an 
electric plant, the capital invested in it would certainly be very 
advantageously employed. 

Heating Effects of the Current. In the transmission 
of electric power it is desirable to have as little of the energy 
transformed into heat as is possible. On the other hand, when 
we want to use the energy as heat we should plan our apparatus 
so as to have the electrical energy liberated in the particular 
limited space where it is wanted. 

It is thus very important to know definitely the relations that 
the number of heat units bear to the numbers of volts, amperes, 
and watts. 



These relations were investigated by James Prescott Joule, who 
made the first determination of the mechanical equivalent of heat. 
Joule placed a small coil of platinum wire in a calorimeter 
with a weighed quantity of water and a thermometer (Fig. 154), 
and in the usual manner (cf. Art. 126) meas- 
ured the quantity of heat given up to the 
water when currents of different strengths 
were passed through the coil. 

Besides measuring the pressure and cur- 
rent strength, he also measured the time 
intervals during which the current had been 
passing. As a result of these experiments, 
which were subsequently repeated with greater 
accuracy by the late Professor Rowland of 
Johns Hopkins University, the following rela- 
tions were established : 

Fig. 154. Joule's 

263. Joule's Law. The number of heat 
units generated by a current of electricity is 
directly proportional: v 

1. To the square of the current strength, 

2. To the resistxince of tJie conductor. 

3. To the time during which the current passes. 

If H represents the number of calories liberated in / seconds, 

C the current strength, and R the resistance, Joule's law may be 

expressed by the equation, — = .24 C^R. Or, 

calories in one second = .24 X (amperes)^ X ohms. (13) 

The factor .24, therefore, represents the number of calories 

per sec. corresponding to one watt. 

This is apparent from Ohm's kw: for C = -^ or E = CR; 

Also, from the equation for electrical power, A = CE; 
so, substituting CR for E, this equation becomes A = G^R; or 
watts = (amperes)^ X ohms. This, when multiplied by .24, 
gives the number of calories in one second as stated by Joule's 
law (c/. equations 11 and 12, Arts. 254 and 256). 


An inspection of these equations, E = CR and A = C^R 

(or — = .24 C^R)y will tell us what to do when we want to transmit 

electrical energy with the least possible loss by heating, and also, 
on the other hand, what to do when our purpose is to get heat 
from the current. 

264. The Heat Loss in Transmission. When considering the 
energy lost in transmission, we note from Ohm's law, E = CR, 
that the drop in voltage, caused by the constant resistance of the 
line, is proportional to the current C. Also from Joule's law, the 
energy dissipated in the line as heat is proportional to C^, There- 
fore we may make these losses small by making the current as 
small as possible. 

Now, the power delivered by the line is measured by the 
product of the current C and the pressure E at which it is deliv- 
ered; and since the electrical power is the same so long as this 
product CE remains constant, we can deliver the same amount 
of it with less line loss by making C smaller and E larger in the 
same proportion. For example, suppose we wish to deliver 1000 
watts to a certain house. We can do this by means of a current 
of 10 amperes at a pressure of 100 volts, or by a current of 5 am- 
peres at a pressure of 200 volts. If the resistance of the line is 2 
ohms, then, in the first case, the power lost in the line by heating 
is, watts = C^R = 10^ X 2 = 200. In the second case the loss is 
watts = 5^ X 2 = 50. Thus, by doubling the voltage, we have 
reduced the watts lost in the line to one-quarter of its former 
value ; and we can understand why it is more economical to trans- 
mit electrical energy at a high voltage and a low amperage. 

If it is desirable to save copper instead of energy, we see that 
we may in the second case get the same efficiency as in the first, 
using a wire of one-fourth the cross-sectional area. For if the 
wire is reduced to one-fourth its former size, its resistance will be 
4 times as great, or 8 ohms. Then the heat loss in the wire is, 
5* X 8 = 200 watts, as at first. 

From these examples we see that by doubling the voltage 
while the total output of the dynamo remains the same, we can 



save either three-fourths of the energy that would be lost in 
transmission, or three-fourths of the copper, as we may elect. 
Thus again it appears that it is mare economical to transmit 
electrical energy at high pressure and small current strength. 

This superior efficiency of high voltage transmission is taken 
advantage of in lighting and power circuits- in some very inter- 
esting ways, two of which we will now consider. 










Fig. 155. The Three-Wire System 

265. The Three-Wire System. Fig. 155 shows a method of 
wiring much used for motors and incandescent lamps. It will 

be seen from the 
diagram, that if 
equal numbers 
of lamps are 
flowing on the 
two branches, the 
current will flow 
through the pairs 
of lamps in series, 
and that a given 
number of lamps 
will take twice the voltage, but only half the current taken by the 
same number of lamps on the two-wire plan. 

As shown in the preceding section, the line loss then will be 
only one-fourth what it would be on the two-wire plan. If the 
two sides are balanced, i.e., if they are equally loaded, no current 
will traverse the middle or neutral wire. The lamps, however, 
are independent of one another and of the motors. This is clearly 
shown by the diagram, for if a lamp b is cut out of the negative 
side, then the current from a can no longer pass through b to the 
negative wire, so it returns by the neutral wire. And if one of the 
lamps a is cut out of the positive side, the current necessary to sup- 
ply its counterpart b, although now no longer fed to b through 
a, is supplied via the neutral wire from the dynamo 7)— on the 
negative side. When the two sides are balanced, the two dyna- 
mos work strictly in series, giving double the voltage of one; 
and the second, 2)—, sends all its current through the first, 2)-h; 



but when the demand is greater on the negative side than on the 
positive, the dynamo D — sends a Sufficient part of its current out 
along the neutral wire, 
and so supplies this 
extra demand. With 
this arrangement, for a 
given number of watts 
delivered and a given 
line loss, the + and — 
feeders can be reduced 
to one-fourth the weight 
required by the two- 
wire plan; and since 
the neutral wire has 
the same cross-section 
as one of the feeders but only half the length (and weight) of 
the two feeders, the total weight of copper used is i + i = i of 
that required by the two-wire i)lan. 

Fig. 156 shows how electrical power is utilized when trans- 
mitted to a large shop. The motor is direct-connected to a 
lathe which is turning axles for car wheels. Figs. 157 and 158 

Fio. 156. Motor-Driven Lathe 

Fig. 157. Motor-Driven Shop 

Fig. 158. Belt- Driven Shop ' 

show the advantage of transmission by wires over transmission by 
belts and shafting. 


266. Alternating Current Transmission. We are now pre- 
pared to appreciate the value of the alternating current trans- 
former (Art. 245), for by means of a transformer a current can, 
with very little loss of energy, be "stepped up" from 125, 250, 
or 500 volts in the generator to 5,000, or 10,000, or even to 50,000 
volts in the feeders, and then ''stepped down" again to 220 or 110 
volts by the means of another transformer. Thus the current 
is at a high voltage during transmission, and the losses in the 
line are reduced to a minimum. These high voltage currents 
are very dangerous and easily escape by leakage unless the con- 
ductors by which they are carried are thoroughly insulated. 

Since alternating currents can be transmitted with so much 
greater economy than can direct currents, the former are rapidly 
displacing the latter for most purposes. 

267. Electrical Heating. Having found in the preceding 
paragraphs that the heating effect of the current increases as the 
resistance and as the square of the current strength, it Is clear 
that when we want to convert the current energy into heat we 
must have either a large current, or a large resistance, or both. 
By thus converting electrical energy into heat, very high tem- 
peratures may be obtained; so that the process is much used 
in electric welding, and in the reduction of ores. 

In the reduction of aluminum, for example, a current of 2,500 
amperes, under a pressure of only about 8 or 9 volts, is passed 
from a large carbon terminal, through the ore in a airbon-lined 
crucible, which forms the other terminal. The reduction of the 
metal from the molten ore is effected partly by the intense heat 
and partly by electrolysis. Carborundum, which is much used 
instead of emery for grinding edge tools, and calcium carbide, 
which is used for producing acetylene gas for lighting purposes, is 
made in electric furnaces. 

For heating suburban cars, and also soldering tools, cooking 
utensils, flatirons, chafing dishes, and even curling irons, current 
at ordinary pressure is passed through coils of highly resisting 
metal, such as iron, German silver, or platinoid. Electric heat- 
ing, as compared with direct heating by burning the fuel without 

!^ .E 3 

Fig. 159. A Divided Circuit 


transformation through a steam engine and dynamo, is altogether 
too expensive to come into general use in cases of this sort, 
but will always be appreciated when small quantities of heat 
are needed, when the greater convenience and cleanliness offset 
the dispropoiiionate cost. 

268. Divided Circuits. In considering the distribution 
of current in parallel conductors, in connection with glow lamps, 
it is found that if the branches 

have equal resistances they C | ^ 

get equal portions of the cur- . y^ 3 S^^^ _^ 

rent. In electrical engineer- 
ing, it is often necessary to 
know the amount of cur- 
rent on each of several 
branches whose resistances 
are not equal; or sometimes 
it is desirable to arrange the resistance of a branch or "shunt" so 
as to switch off a definite fraction of the current. 

We must, therefore, know how the resistances of the branches 
govern the distribution of the current among them. In order to 
find this relation, let us consider the conductor. Fig. 159. 

It is evident from this diagram that there is a definite differ- 
ence of potential, or loss of pressure, between A and B, and that 
this must be the same for each branch. 

Let us call this difference of potential E volts. 

E volts 
The current on AC.B is then C, = ^ , , or E = C, X 2, 
^ ^ 2 ohms 

E volts 
6 ohms' 
the difference of potential is the same for both branches, 

c,x2 = c,x ^ = ^«r§^ = | = r 

Of the 8 amperes, therefore, the 2 ohm branch gets 6 am- 
peres or f , and the 6 ohm branch will get 2 amperes or J, i.e., the 
current strength on each branch is inversehj propoi-tional to the 
resistance of that branch. It may be proved, both mathematically 

and the current on AC2B is Cj = ^ 1 ,„ > or £ = C2 X 6. Since 



and experimentally, that this is true for any number of branches 

Shunts. We may apply this principle when we are 
using a delicate galvanometer with a strong current, and wish to 
send only xu^inr of the current through it, so as not to injure it. 
We then connect a shunt across its terminals, and the resistance 
of the shunt must be j^-^ of the resistance of the galvanometer 
branch. The current will divide between the galvanometer coil 
and the shunt as it does between AC^B and AC^B, Fig. 159; 
so the galvanometer will then get j^Vtt of the current and the 
shunt the remainder, or iVoV- 


270. Arc Lamp Eegnlation. Another very important appli- 
cation of the shunt principle is found in the regulating magnets 
of the arc lamp. 

The diagram. Fig. 160, shows on the reg- 
ulating magnet two windings, which are car- 
ried around it in opposite directions, one of a 
few turns of thick wire and in series with 
the carbons, and the other a shunt coil of 
many turns of fine wire. When the current is 
turned into the lamp, the carbons are in 
contact, and their resistance is small; so that 
the current is cori-espondingly large. This cur- 
rent, going around the series coil, magnetizes 
it strongly. This causes it to pull up the core 
C, which in its turn acts on the clutch K so 
as to pull up the + carbon. When the + 
carbon is pulled up too far, the current 
through the carbons is reduced, because of the 
greater resistance of the lengthened arc; but 
because of this increased resistance a greater 
proportion of the supply current'goes around the shunt coil. This, 
being wound in the opposite direction, counteracts the magnetic 
pull of the series coil and lets the clutch down, so that the car- 
bon slips through it and the arc is shortened. If the arc gets 



FiQ. 160. The Arc 
Lamp Regulator 



too short, the diminished resistance allows more of the current to 
go through the series coil* and less through the shunt coil, so 

the series coil pulls the clutch 
up and lengthens the arc. 

If the resistances of the 
two coils are properly pro- 
portioned, the carbons will 
be kept a constant distance 
apart, as mentioned in Art. 
240. This principle of di- 
vided circuits has important 
applications in the shunt 
windings of the field mag- 
nets for dynamos and motors, and in connection with controllers 
used in starting motors. 

Fig. 161. Section of a Lifting Magnet 

271. Lifting Magnets. Another important application of elec- 
tro magnets remains to be considered. This is the lifting magnet. 
Fig. 161 represents a cross-section 
of one, showing how the poles are 
arranged so as to have a conaplete 
magnetic circuit. Anyone who has 
seen one of these magnets in opera- 
tion, lifting heavy steel plates or gird- 
ers and carrying them about a shop, 
will appreciate their convenience. 
Fig. 162 is a photograph of such a 
magnet holding a mass of iron that 
weighs about 5 tons. This mass of 
iron is called a "skull cracker,*' and 
it is used to break up old castings 
before remelting them. The mass is 
lifted by means of the magnet and 

then dropped on the castings. Fig. 162. The Lifting Magnet 

AT Work 

272. Voltaic Cells. We have found the answers to most of 
the questions that were raised at the beginning of this chapter, 


c z 


FiQ. 163. Batteries i^ Series 


and we shall now try to answer the others. Of the discovery of 
the voltaic cell and the important fact that it supplies a continuous 
current of electricity, we learned something in Chapter XI. But 
we were then interested chiefly in the discoveries in electromag- 
netism, which immediately followed the discovery of currents, 
and so we deferred the study of voltaic and electrolytic action 
until now. 

Battery cells, as well as other electric generators, must have a 
COMPLETE CONDUCTING CIRCUIT in order to do work. Fig. 163 

shows a good way of repre- 
senting a circuit in which, for 
example, an electromagnet 
is operated by several cells 
in series (c/. Art. 258 and 
Fig. 152). In such a case it 
is found that both the volt- 
ages and the resistances of 
the circuit are added up, which is what we should expect, since the 
current goes through one cell after another. It follows, therefore, 
that if we have any number of equal cells in series, with some 
external resistance: 

(1) The total electrcmiotive force of the circuit is that of a single 
cell multiplied by the number of cells. 

(2) The resistance of the battery itself, which is called the in- 
ternal resistance, is that of a single cell multiplied by the number 
of cells. 

(3) The total resistance of the circuit is the internal resistance 
plu^s the sum of all the external resistances. 

(4) The current strength, therefore, may he found by dividing 
the total number of volts E. M. F. by the total number of ohms 

273. Energy of the Cell. When a voltaic cell is sending a 
current around a circuit, it will be noticed that chemical changes 
are going on. Bubbles of hydrogen gas are seen to be Hberated 
at the copper plate. Also, if we weigh the two plates before 
operating the circuit, and after some time remove them and 


weigh them again, we shall find that the zinc plate has diminished 
in mass, while the copper plate has not. 

If we were to determine the chemical composition of the fluid, 
we should find less sulphuric acid in it than before, and we should 
find a new compound, zinc sulphate. The relations between the 
quantities of the substances taking part in this chemical change 
are found to be constant. Thus, for every 65 grams of zinc that 
disappear, 98 grams of sulphuric acid are used up. At the same 
time 2 grams of hydrogen are liberated, and 161 grams of zinc 
sulphate are made. These relations may be represented by 
symbols, as follows : 

and produce and 

Zinc Sulphuric acid Zinc Sulphate Hydrogen 

Zn +H2SO, =ZnSO, H^ 

65 gm 2+32+64=98 gm 65+32+64=161 gm 2 gm 

The s3mibols, S and O^ represent sulphur and oxygen in the 
proportions of 32 and 64 gms, respectively, to 2 of hydrogen or 
65 of zinc. From an inspection of the relative weights of the 
substances, it is evident that the zinc replaces the hydrogen in 
the sulphuric acid and combines with the SO4 (or sulphion, as 
it may be called). 

Now, we know that we get energy from carbon by burning it, 
i.e., causing it to combine with oxygen from the air. Similarly, 
it ought now to be quite plain that we get energy from the voltaic 
cell by causing zinc to combine with the oxygen in the sulphine of 
sulphuric acid, as was stated in Art. 229. 

Thus chemical energy is transformed into electrical energy. 

274. Polarization of a Voltaic Cell. A very troublesome 
defect in Volta's cell is that the hydrogen bubbles stick to the 
copper plate and diminish the current in two ways: 

1. Hydrogen is a bad conductor, as all gases are, and it insu- 
lates that part of the plate which it covers, so the internal resist- 
ance is increased. 

2. It tends to recombine with the acid and displace the zinc. 
This tendency gives rise to an electromotive force which tries to 
send a current backwards through the cell. This counter electro- 



motive force weakens the current, 
cell is said to be polarized. 

When in this condition, the 

275. Why is the Hydrogen Liberated at the Copper Plate ? 
We have seen that the metal which combines with the SO4 and 
releases the hydrogen is the zinc, not the copper; for the zinc is 
found to be used up, while the copper is not. It seems, then, 
that although the chemical action starts at the zinc plate, yet the 
copper plate is the one at which the hydrogen is actually liberated. 
In order to explain this, a hypothesis has been advanced, which 
will be understood by reference to Fig. 164. 

It is supposed that when the sulphuric acid is dissolved in the 
water, some of its molecules are always broken up into parts, 

each of which carries a 
charge of electricity, one 
kind of part being positively 
charged hydrogen; the other 
kind, negatively charged sul- 

It is conceived that the 
solution is in an unstable 
condition. Some of the 
molecules arc always broken up into these electrically charged 
parts, or ions, as they are called, and the ions are contin- 
ually recombining to form molecules. If this is really the case, 
it is easy to see that as a result of electrical attractions between 
the oppositely charged ions, a series of combinations would take 
place, as indicated by the arrows in the diagram, and there would 
be a procession of positively charged hydrogen ions toward the 
copper, and of negatively charged sulphions toward the zinc. 

276. The Ion Hypothesis. This hypothesis not only gives us an 
idea as to why the hydrogen comes off at the zinc, but also gives us 
some conception of a mechanism by which electricity may be con- 
ducted through a liquid and how the liquid may be broken up while 
conducting the current. It is also found to fit well with the molec- 
ular hypothesis, which we found it convenient to adopt while 

Fio. 164. The Ions Change Partners 



studying heat, for it explains many phenomena of solutions, 
which would be very difficult to understand otherwise. 

The plate toward which the + ions travel, the copper in this 
case, is called the cathode, and the one toward which the — 
ions travel, the zinc in this case, is called the anode. The direc- 
tion of the current is conceived to he from anode to cathode through 
the liquidy and from cathode to anode along the imre. 

277. Local Action. Another defect of the simple voltaic 
cell arises from the fact that the zinc continues to waste away, 
even when the circuit is open, so that no current is passing. 
This waste may be almost entirely prevented 

by amalgamating the zinc with mercury. 

The reason for this will be understood from 
Fig. 165. Millions of little particles of carbon 
and iron exist as impurities in the plate, 
and, with the neighboring zinc particles, they 
form diminutive voltaic cells which give risu 
to little local currents, as represented by the 
arrows in the diagram. As these little cur- 
rents travel in short circuits, and never get out 
to the conducting wires, all their energy is 
transformed into heat and is wasted. 

The mercury prevents this local action 
by dissolving zinc to form a zinc amalgam, which Spreads over 
the zinc plate and covers up the impurities. 

278. Commercial Cells. Very early in the history of the 
simple voltaic cell, it was found to be inefficient. Accordingly, 
several modifications have been made to increase the output. 

1. Since the voltage depends on what substances are used tor 
the plates and the active fluid, various metals and fluids have 
been tried. Carbon or copper is now used almost exclusively 
for the anode, zinc for the cathode, and either sulphuric acid 
or ammonium chloride (sal ammoniac) for the active fluid. 

2. The internal resistance is diminished (a) by giving the 
plates as large a surface as ])ossible, and (b) by diminishing the 
distance between them? Why? 





Fig. 165. Local 



3. Polarization is remedied by using some oxidizing agent, 
i.e., a substance that easily breaks down chemically and gives 
off oxygen, which combines with the hydrogen to form molecules 
of water. Polarization is prevented by introducing a second 
fluid, which deposits a metallic ion instead of hydrogen. 

279. The Chromic Acid Cell. This is the best kind of cell 
for amateur use and for schools that are not equipped with a 
direct current dynamo, or storage battery. The anode is carbon, 
the cathode zinc, the active fluid sulphuric acid, and the depolar- 
izing agent chromic acid. 

280. The Leclanche Cell. Many forms of this cell are sold 
under various trade names. The anode is carbon, the cathode 
zinc, the active fluid ammonium chloride, and the depolarizing 

substance a black powder, called manga- 
nese dioxide. This is compacted around 
the carbon, or inclosed with it in a porous 
cup of either carbon or earthenware. This 
type of cell polarizes very rapidly in spite 
of the action of the manganese dioxide, 
and hence should never be used in a cir- 
cuit that is to be kept closed for any consid- 
erable time. It is a very good cell for 
door bells, electric gas lighting, local bat- 
tery for telephone transmitters, etc., when 
the current is used during short aiid in- 
frequent intervals. The so-called dry cells 
are of this type. 

Fio. 166. Gravity Cell 

281. The Gravity Cell. This is a two- 
fluid cell. The anode is zinc, and is placed 
at the top of the cell; the cathode is copper, 
and is placed at the bottom. The active fluid is very dilute sul- 
phuric acid, and floats on the depolarizing fluid (a solution of 
copper sulphate), which is heavier than the sulphuric acid, and so 
remains at the bottom. Polarization can not occur when the cell 


is in good condition, as will be seen by reference to Fig. 166. This 
diagram shows that since the copper cathode is surrounded by 
a copper sulphate solution, there are no hydrogen ions to be 
liberated there. Copper ions are liberated instead, and these 
can do no harm. 

Gravity cells must be kept in a closed circuit with a large 
external resistance, and if the zincs are large and an excess of 
crystallized copper sulphate is kept in them, they last for a long 
time and without attention, and give a very steady current. 
They are in common use for telegraphic work, except on long 
distance lines, where dynamos are now used. 

282. Electrolysis. In the voltaic cell, ions combine and give 
up their charges, producing an electric current. This process is a 
reversible one, for if a current from a dynamo or voltaic battery 
be sent between two plates of metal immersed in a solution that 
contains a salt of any metal, the ions of this salt immediately begin 
to progress in opposite directions, the metallic or positively charged 
ions going toward the cathode, and the non-metallic, or negatively 
charged ions, going to the anode. 

This process of breaking up a liquid compound by passing a 
current through it, is called electrolysis (c/. Art. 208). 

The liquid is called an electrolyte, or electrolytic con- 
ductor. The anode and cathode plates are called electrodes. 
The relations which exist in connection with electrolysis were first 
thoroughly investigated by Sir Humphry Davy, who discovered 
metallic sodium and potassium by this process, and by means of 
it, made many important advances in chemical science. Davy's 
work was followed up and greatly extended by Faraday, who 
discovered and announced the following relations: 

283. Faraday's Laws of Eleotrolysis. 1. The amount of 
chemical action is the same in all parts of the circuit. 

2. The mass of any kind of ion liberated at an electrode is pro- 
portional to the quantity of electricity that passes, i.e., to the product, 
amperes X seconds. 

3. The mass of any kind of ion liberated by a given quantity 
of electricity is proportional to its chemical equivalent, i.e., to the 



number of grams of it that combine with a gram of hydrogen (or 
replace one gram of hydrogen in combination). 

Knowledge of the laws of electrolysis has been employed in 
many ways, some of the most interesting of which will be described 
presently, but none of its applications is so important as its use 
as a means of investigation in theoretical chemistry, upon which 
all practical chemistry is based. Among the other uses of elec- 
trolysis we may mention the measurement of current strength 
for the purpose of standardizing galvanometers, voltmeters, and 
ammeters, the reduction of metals from their ores, the refining 
of crude copper for electric conducting wires, the making of 
electrotype plates from which books are printed, electroplating, 
and the storage battery. 

284. Electroplating. Fig. 167 shows an electroplating bath. 
It is usually a large vat, containing a solution of some compound 

of the metal which is to be 

deposited. Thick copper 
rods, or "bus bars," are 
laid along its length, and 
two of these, B+ B+, are 
connected with the + 
feeder from a dynamo, the 
other with the — feeder. 
From the + bus bars 
are suspended, as anodes, large plates A A of the metal to be depos- 
ited, and from the — bus bar are hung, as cathodes KK, the arti- 
cles to be plated. When the current is passed, the procession 
of metallic ions goes toward the cathodes, giving them the desired 
coating, while for every ion deposited on the cathode, an ion from 
the anode goes into the electrolyte and takes its place. 

The electrolyte thus remains constant in strength, while the 
anodes lose as much metal as the cathodes gain. 

Fig. 167. An Ei-ectroplating Bath 

285. Storage Batteries. The storage battery is the converse 
of the electroplating cell; the electrolyte is dilute sulphuric acid 
and the electrodes are perforated lead plates, photographs of 



which are shown in Figs. 168 and 169. The perforations 
in these plates are filled with oxides of lead. When a current 
from a dynamo is passed through this cell, hydrogen ions go to 
the cathode and reduce the lead oxide there to metallic lead. 
Simultaneously, sulphions go to the anode and there give up oxy- 
gen, which combines with the lead oxide to form peroxide of 
lead, i.e., an oxide with a greater proportion of oxygen in it. 
When all the oxides are thus changed, the cell is said to be fully 
charged; and this condition is indicated by a copious evolution of 
hydrogen and oxygen gases at the electrodes. This charged cell 
is in a highly polarized condition (c/. Art. 274); it has a counter 

FiQ. 168. Storage Battery, Pos- 
itive Plate 

Fig. 169. Storage Battery, Neg- 
ative Plate 

E. M. F. of polarization equal to about 2 volts, which tends to 
send out a reverse current. 

In fact, if 55 of these cells in series are charged by a small 
110 volt dynamo, and the belt by which the dynamo armature is 
driven be thrown off, this reverse current will go back through 
the dynamo and run it as a motor. The storage battery, theriy is 
a battery in which electrical energy w transformed into chemical 
potential energy and stored np for future use. The reversed pro- 
cession of the ions begins whenever the electrodes are joined 
through an external conductor, and the chemical energy is re- 
converted into electrical energy, ready to do any kind of work. 



Fig. 170. Storage Battery Plant 

Storage batteries are used for running automobiles, but their 
greatest use at present, is in large power plants, where they are 
employed to store energy when the demand is light and pay it out 

again when the demand 
is extra heavy. Fig. 
170 shows such a stor- 
age battery, belonging 
to a large power plant. 
They have not yet come 
into very extensive use 
for automobiles because 
of their great weight 
and liability to dete- 
rioration when not 
properly cared for. 
There ought, however, 
to be a great future for 
them in connection with windmills, because energy could be stored 
up in them when the wind was blowing strongly and taken from 
them when the wind was too light to operate the mill. 

286. Eetrospect. Before leaving the study of electricity and 
magnetism, it may be useful to review some of the important 
things we have learned about them. We have discovered that 
when two bodies, made of different substances, are brought in 
contact and then separated, each is electrically charged, one posi- 
tively, and the other negatively. These electrically charged bodies 
have been found to act on other bodies not in contact with them; 
and this action takes place not only through dielectrics, but also 
through a vacuum. We have seen that a magnet acts on another, 
or on a magnetic substance, although air, or wood, or other sub- 
stances are between them. We have learned how an electric cur- 
rent is generated by chemical action in a voltaic cell, and how 
such a current is equivalent to electrostatically charged particles in 
rapid motion. Finally, we found that a magnetic field may be 
obtained by passing a current through a wire or coil ; and con- 
versely, that a current may be produced by moving a magnetic 


field or changing its strength in the neighborhood of a closed 
conducting circuit. 

All of these effects, produced by charged bodies, currents, and 
magnets, take place without apparent connection between the 
bodies that so act. Since it is hardly conceivable that these actions 
are effected without any connection whatever, we are constrained 
to assume that they are produced by stresses in some medium. 
If we have to suppose that a medium exists for these phenomena, 
is it not simpler to conceive that this medium and the one that 
transmits the heat waves are the same? Let us then adopt the 
hypothesis that these electric and magnetic phenomena are mani- 
festations of stresses of some kind in the ether. We shall have 
occasion to recall this h3^othesis before we finish the study of 
the other branches of this subject. 


1. Current strength is measured in amperes, electromotive 
force in volts, and electrical resistance in ohms. 

2. The resistivity of a substance is the resistance at 0°,C. of a 
conductor of the substance having unit length and unit cross-sec- 
tional area. 

3. The electrical resistance of any substance may be found 
from the equation: 

Resistance _ Resistivity X Length 
Area of Cross-section 

4. Ohm's Law is expressed by the equation: 


Amperes = -tt-, . 

^ Ohms 

5. Joule's Law. The power consumed in an electrical con- 
ductor is found by the equation : 

Watts = (Amperes)* X Ohms = Volts X Amperes: the heat 
developed, by the equation: 

Calories = Watts X .24. 

6. One horse-power = 746 watts, or 746 X 10^ |^. 

7. Electricity is distributed in series and in parallel circuits; 
sometimes by a combination of both. 


8. Economy of electrical transmission is secured by using high 
electrical pressures. 

9. Electrical power is utilized by means of motors. Elec- 
trical heating is done by means of coils of high resistance, or by 
suitable electric furnaces in which the heating action is somewhat 
similar to that of the arc light. 

10. In divided circuits, the current in each branch or "shunt,'' 
is inversely proportional to the corresponding resistance. 

11. Delicate electrical apparatus may be used with large 
currents if suitable shunts are employed. 

12. In a voltaic or electrolytic cell the electricity is conducted 
by a progression of + ions toward the cathode and of — ions toward 
the anode. 

13. The polarization of a voltaic cell may be remedied by 
oxidizing the hydrogen as in the chromic acid or the Leclanch^ 
type; or it may be prevented by employing a second solution 
from which are deposited ions of the cathode metal^ as in the 
gravity type. 

14. ,When cells or other electric generators are joined in 
series, the total, voltage is the sum of the voltages of all, and the 
total resistance is the sum of all the resistances of the circuit. 

15. Faraday's Laws of electrolysis state that the mass of an 
ion liberated is equal to its electro-chemical equivalent multiplied 
by the product of the current strength and the time. 

16. The most common and important applications of elec- 
trolysis are: 1. Current measurement. 2. Electroplating and 
electrotyping. 3. Reduction of metals from their ores, and the 
refining of crude metals. 4. Storage batteries. 


1. How many amperes are there in a current which deposits 
16.77 gm silver in 50 minutes? How much silver will be deposited 
by a 2 ampere current in 30 minutes? 

2. What electromotive force will send a current of 10 amperes 
through a lamp whose resistance is 4.8 ohms? What is the resistance 
of an arc lamp which takes a current of 15 amperes at a pressure of 
65 volts? 

3. What is the rate (watts) at which each of the lamps of problem 


2 consumes energy? Find the mechanical equivalent (horse-power 
hours) of the energy used by each in 12 hours. 

4. Find the voltage and the power (K. W.) of a dynamo that 
will operate a series arc lamp circuit, the data being as follows: 
Number of lamps, 25; volts for -each lamp, 45; current strength, 10 
amperes; length of line circuit 2500 ft.; resistance of wire (ohms per 
thousand ft.)f 2.5. What horse-power must the engine supply to the 
dynamo, if we allow an efficiency of 80 per cent for the dynamo and 

5. How many gm cal of heat may be obtained from a current of 
2000 amperes at a pressure of 12 volts? How many Kg of copper 
may be raised from 14° C. to its melting point (1054°) by this heat 
if the mean specific ht. of copper for this temperature range is 0.105? 

6. The following problem illustrates the way in which automobile 
engines are tested in a certain large factory. A car was lifted on jack 
screws, and the driving wheels were belted to a dynamo. When 
the angular velocity of the drivers corresponded to a linear car speed 
of 25 ^^-^ the dynamo was able to light 128 16-candle-power glow 
lamps connected in parallel circuit. The meters showed that the 
lamps were taking half an ampere each at 110 volts pressure. They 
were so near the dynamo that there was no line drop. What were 
the total current strength and the electromotive force of the dynamo? 
Its output (watts)? What H.P. did it take from the engine, allowing 
for 25 per cent loss in the dynamo and belt? The answer corre- 
sponds to the horse-power developed by the engine when the car has 
the given speed on a smooth, level road. 

7. A waterfall 6.10 m high delivers 15,000 Kg water per minute. 
What is its H.?.? What H.P. may be delivered from it by a water- 
wheel having an efficiency of 70 per cent through a dynamo having 
an efficiency of 85 per cent? The output of this dynamo will equal 
how many watts? If its electromotive force is 125 volts, what 
current will it supply? 

8. The same fall, measured in foot and pound units, gives; pounds 
of water per minute 33,000, height 20 ft. Answer the questions 
and compare the answers with those of question 7. 

9. The dynamo, problems 7 and 8, delivers its current to a num- 
ber of glow lamps in parallel, with a 10 per cent drop in voltage 
through the feeders. What is the voltage through the lamps? The 
current delivered by the feeders? Allowing 0.5 amperes per lamp, 
how many lamps may be operated? Calculate the line resistance 
and the watts lost in the line, from the data here given. If the line 
is 500 feet long, consult the wiring table, p. 298, and find the gauge 
number of the proper size of wire to use? 

10. Suppose in the lamp circuit, problem 9, a pair of branch 


wires have a length of 100 ft. and supply 20 lamps, what current 
must they carry? They are to be chosen so as to have a 2.5 volt 
drop. What is their resistance? From the wiring table find the 
gauge number to be selected. Find the rate (watts) at which 
energy is lost in these branch wires, and in the feeders. Find the 
heat developed (gm cal) in each case. 

11. An electric bell has a resistance of 10 ohms and works per- 
fectly when connected with short wires to one dry cell having an 
electromotive force of 1.5 volts, and an internal resistance of 0.3 
ohm. What current is it then using? The same bell is connected 
on a door bell circuit of 150 ft. of number 20 copper bell wire, and 
although the current is found to be complete, it does not work. 
Mention two ways in which the trouble might easily be remedied. 
From the wiring table find the resistance of the line wire and again 
calculate the current strength. Suppose two more cells just like 
the one in the last problem were connected in series with it and the 
circuit. Calculate the resulting current. 

12. In table II, p. 299, the resistivities of some metals are given, 
the resistivity for this table being defined as the resistivity in ohms 
of a conductor one meter long and one square millimeter (1 mm') 
in cross-sectional area. From this table and the laws of resistance 
(Art. 253), find the resistances of the wires for which the following data 
are given: 1. A silver wire 3 m long and O.l mm in diameter. 2. 
A German silver wire 20 m long and 0.5 mm' in cross-section. 3. A 
lead wire 0.6 m long and 0.8 mm' in section. 

13. From the table of resistivities express the resistances of each 
of the materials in terms of the resistance of copper as a unit; thus, 
other things being equal, the resistance of a platinum conductor is 
how many times that of a copper one? 

14. A galvanometer has a resistance of 5 ohms, and it is desired 
to use it with a current of 10 amperes; but the greatest current that can 
be sent through it without either injuring it or causing its deflections 
to be too great to read is 0.1 ampere. What must be the resistance of 
a shunt that will produce the desired result when connected across 
its terminals? 

15. In Fig. 171 a current divides at A and reunites at B. A gal- 
vanometer G is connected across, as shown. If the current on ACxB 
is Cj, and that on A C2 B is C72, show that the pressure on the part 
A Ci is ei = CiTif and that on the part A C2 it is 63 =C2r3. Now, if 
the point of contact C2 be moved along A C2 B toward A or B till a 
place is found such that no deflection of the galvanometer occurs, 
show that under this condition the points Ci and C2 are equi-potential; 
i.e., t'lere is no difference in electrical pressure at these two points. 
Prove that when thb is the case 63 = Cg, hence CiTi =5 C2rz (o). In a 


similar way, prove that under the same conditions Cir2 = C2r4 (&). 
Divide the equation (a) by the equation (6) and simplify. What 
relation has been proved to exist among the four resistances, when no 
current passes through the galvanometer? 

16. Suppose in the last probleni the resistances r2,r3,r4 can be varied 
at will, and that r^ is a certain unknown resistance whose value we 
wish to find. For example, we 

make ra = 1 ohm, r^ = 100 ohms, 
Ti the unknown, and then begin to 
vary r2 until we have found a 
value for it which will leave the 
four resistances so adjusted that 
no current passes through the gal- 
vanometer. Now Ti is the only 
unknown quantity, and the equa- Fig. 171 

tion obtained in the previous 

•problem applies, for the conditions are the same as there supposed. 
What is the numerical value of ri when r2 = 250? 

17. Show that if AC B is made of a wire of uniform cross-sectional 
area and material, the ratio of the lengths of the two parts A C2 and 
C2B may be substituted for the ratio of the actual resistances, and 
give the same numerical value as before for the unknown resist- 
ance. Apparatus arranged to measure electrical resistance in the 
ways suggested by problems 15-17 is called a Wheatstone bridge, and is 
of great service in practical electrical work as illustrated in this chapter. 

18. Does the electromagnet, Figs. 161, 162, really do any "lifting," 
or does it simply hold while the crane-hobt to which it is attached 
does the lifting? Then does such a magnet use a great amount of energy? 


1. If you are permanently interested in electricity or expect to go 
farther in the study of it than this course can take you, you ought to 
own these, two books; Elementary Lessons in Electricity and Magnetism, 
by S. P. Thompson, and Elementary Electricity and Magnetism, by D. C. 
and J. P. Jackson (both published by Macmillan, New York). 

2. Find out from books, or the bulletins of electrical manufacturers, 
or from some friend who is an electrician, what a "booster" or rotary 
transformer is, and how it is useful in connection with a storage battery 
for an electric railway power plant, or an electric light station. If you 
can get the information, write a short paper about it. 

3. From similar sources, get information about the use of a starting 
resistance with a shunt motor, as shown at -B, Fig. 135, and put the 
results into the form of a written report. Can you find out anything 
about the counter-electromotive force of a motor in this connection? 



About the danger of a "field discharge" when a shunt motor is sud- 
denly shut off, without having in circuit with the armature a suitable 
starting resistance like that mentioned? See Practical Electricity, 
by J. C. Lincoln (published by the Cleveland Armature Works, Cleve- 
land, Ohio), which is excellent on the subjects treated in this chapter. 

4. Find out also how the electromotive force of the shunt coils 
of a shunt or compound woimd dynamo may be regulated for varying 
loads by means of a variable resistance, placed in the same branch of 
the circuit with the shunt coils of the field magnets, as in Fig. 150. 

5. Visit an electrotype foundry or electroplating shop, find out 
what you can and make a report. 

6. Find out what you can about electric elevators and hoists. 

Characteristics op Copper Wire 









Num. • 


Brown & 


Imil = 



Mils. d2. 


1000 ft. 

Ohms per 

1000 ft. 

at 75» Fahr. 

= 24«C. 




1mm = 
0.001 cm 






















































































































































Wire. • 


















The following table gives the resistivities of some metals, the re- 
sistivity for this table being defined as the resistance at 0° C of a con- 
ductor of the metal having a length of one meter and a cross sectional 
area of one square millimeter. The resistivity of German silver and 
other alloys varies with the composition. 


Ck)pper 0.017 

Silver 0.016 

Platinum 0.108 

Lead 0.210 

Iron 0.100 

German silver . 34 

Practical Use of Tables I and II 

Electric wiremen often use a formula for determining the cross 
section of a wire of given dimensions and having a required resist- 
ance, and express this cross section in circular^ mils. A wire of cir- 
cular cross section and one one-thousandth of an inch (1 mil) in 
diameter is said to have a cross section of one circular mil. They 
also define the specific resistance of a wire as the resistance of one 
mil-foot (i.e. 1 mil in diameter and 1 ft. long) at 75® Fah, The re- 
sistance of a mil-foot of good commercial copper wire is 10.5 ohms. 
If L represents the length of a wire in feet, CM its. cross section in 

circular mils, show that its resistance R=10.5 77^7 ohms. Also that 
* CM 

OM = — ^^— ^. Wiremen also determine the cross section of a wire 
of a certain length, which wUl cause a given drop with a given cur- 
rent by the equation, CM= — —y wherein CM represents cir- 
cular mils, L the length in feet, A the amperes carried, and V the 
drop in volts. Show from the preceding formula and from Ohm's law 
that this one is correct. 

From these formulas and tables I and II you can make up and 
solve as many wiring problems of this sort as you like. 

Another good exercise will be to find the specific resistances in 
mil-foot units of the substances in table II by multiplying the re- 
sistance of a mil-foot of copper (i.e. 10.5) by each of the ratios ob- 
tained in problem 13, p. 296. 

If you are taking the commercial course of your school, an inter- 
esting exercise will be to calculate the cost of the copper, also to 
get data from a friend who is an electrical contractor and draw up 
specifications and estimates for the two electrical plants describ^.d in 
this chapter. 



287. Of Waves. In Chapter VIII, while studying the trans- 
fer of heat from one body to another when there is no visible 
contact between them, we were led to adopt the hypothesb that 
heat energy is propagated across apparently empty space by some 
form of invisible wave motion. Similar reasons lead us to adopt 
a wave hypothesis to describe the phenomena of sound and light, 
and we can best appreciate how beautifully this theory fits all the 
facts together into an intelligible story, if we first make a little 
study of the waves with which we are al! /amiliar. Let us, then, 
ask: How do waves originate? What properties must a me- 
dium possess in order to be capable of transmitting waves? What 
is the mechanism by which they are propagated? 

288. Water Waves. We are all familiar with water waves; 
for who has not amused himself by throwing stones into still 
water and watching the charmingly symmetrical figures produced 
on its surface? There are few of us who are not in possession of 
some vivid mental pictures that may aid us in studying this most 
fascinating portion of our subject. 

9. Origin of Waves. If we throw a stone into a pond and 
watch closely when it strikes the water, we notice that its first 
effect is to push aside the water at the place where it falls. This 
action lowers the level of the water surface, thus leaving a hollow 
behind the stone as it sinks from sight. Since the free surface of 
a liquid always strives, as it were, to remain level, the water, which 
has been thus thrust aside, rushes back as if to fill the 
hollow left by the stone. But when the particles rush in from 
all sides behind the sinking stone, they acquire kinetic energy, 
which carries them further than they intend. The result is that 




water is now piled up in a little heap over the place where the 
stone fell. / 

The water particles then hastily retrace their steps, but again 
they are irresistibly carried past the position they desire to occupy, 
and again a hollow is formed in the surface of the pond; but this 
time it is not so deep as before. This process of piling up and 
retreating is repeated several times, the elevation being less 
marked each time, until the motion at the point where the stone 
struck ceases altogether. A back and forth motion of this sort 
is called a vibratory motion. We thus reach the conclusion 
that waves originate at a point lohere a vibratory motion is forced on 
the medium by some outside body, 

290. Characteristics of Waves. Are the waves that spread 
out on the pond different when produced by a large stone from what 

Fig. 172. Waves From a Large 

Fig. 173. 

Waves From a 

they are when produced by a small one? On trying the experi- 
ment, we find that the waves caused by a large stone are larger 
than those produced by a small one. If we wish to compare them, 
we must agree on a method of measuring them. In order to 
appreciate what the characteristics of waves are, let us suppose 
that a succession of these water Vaves is suddenly frozen solid; 
the shape of the surface resembles the curve in Fig. 174. Exami- 
nation of this curve shows that some parts of the wave are above 
the normal level of the water, while other parts are below it. The 
parts above the normal level are called- crests, those below it are 
called TROUGHS. The distance of the top of the crest from the 


normal level is equal to the distance of the bottom of a trough 

from the same level and is called the amplitude of the wave. 

The WAVE-LENGTH is the distance between two successive crests, 

or between two successive troughs. 

Applying these definitions to the cases of the large and small 

stone^ we see that the waves started by the large stone have both 

a greater amplitude and a greater 
wave length than those about 

Fig. 174. Shape of a Simple Wave ^^^ ^^^^^ ^^^^ ^hus there is 

a relation between the nature of the vibration that started the 
waves and the characteristics of the waves, so that we can judge 
of the vibration by observing the waves. 

291. What Waves Can Tell Us. We have just learned that 
we can form some idea of the magnitude of the disturbance. that 
started the waves by noting the wave length and the amplitude of 
the wave. Furthermore, we can form some idea of the direction 
in which the point of disturbance lies I^y noting the direction in 
which the waves are traveling. Finally, v/e can .infer something 
about the nature of the disturbance from the shape of the waves. 
Hence we see that waves may bring us four kinds of information 
concerning the source of the vibrations, viz.: 1. The direction in 
which the vxLves are traveling indicatec the direction of the source. 
2. The length of the waves informs its as to the rapidity of the 
vibration, 3. The amplitude of the waves tells us of the violence 
of the disturbance, 4. The shape of the wave allows UrS to infer 
something concerning the nature of the vibrations of the source. 

Wave Motion. Another important fact may be learned 
from watching the water waves. If we throw a small chip on 
the surface of the pond, well out from the shore, and observe its 
motion when the waves pass it, we see that the chip is not carried 
along in the direction in which the waves move, but that it merely 
rises and falls while the wave motion passes beneath it. Since 
the chip indicates the motion of the water particles about it, we 
may conclude that the water does not move forward with the wave, 
but merely rises and falls as the wave passes. 



900 00 # 000 

Fig. 175. Particle 1 has Executed i vibration 

This fact leads us to an important conception as to the mechan- 
ism of wave motion. Thus, let us consider a row of particles 
held together by cohesion or some other elastic force (Fig. 175). 
If the first particle is 
displaced in a direction 
perpendicular to the 
rov/, the force that holds 
the two together will 
compel the second par- 
ticle to follow. But since 
the connection between 

the two particles is elastic, not rigid, the second will always lag a 
little behind the first. Hence, when the first particle has reached 
the end of its trip, the second will not have traveled quite so far, 
the third will lag a little behind the second, and so on. There- 
fore the condition of the row of particles, when the first one has 
rpached its position of greatest displacement, will be that shown 
in Fig. 175. 

Particle 1, having reached its position of greatest displace- 
ment, pauses there for a brief instant and then begins to retrace 
its steps. While this particle^is stationary, 2 catches up with it 
and reaches its position of greatest displacement as 1 starts down- 
ward. Particle 3 follows 2 in the same way, and so on. Thus we 
see that the successive particles reach their positions of greatest 
displacement one after another. We may say that this position 
of greatest displacement is passed along from one particle to 
the next. But the position of greatest displacement constitutes 

the crest of the wave, 
and so we get a concep- 
tion of the mechanism 
by which waves are 
propagated along a row 
of particles that are 
held together by cohe- 
sion, or any other elastic force. The positions of the particles 
when number 1 has returned to the starting point are shown in 
Fig. 176. 

Fig. 176. 1 has Executed i Vibration 



Fia. 177. 1 HAS Executed J Vibration 

Now, when particle 1 reaches the position from which it started, 
i.e., when it has completed half a vibration, it is moving with 
considerable velocity. It therefore possesses kinetic energy. 

This energy will cause 
it to move past its origi- 
nal position and to 
make an excursion on 
the opposite side. Hence 
it will now move down- 
ward, dragging the ad- 
jacent particle after it, will reach a position of greatest negative 
displacement (Fig. 177), and return again to the starting point. 
The positions of the particles, when this has been done, are 
shown in Fig. 178. Particle 1 is now in the same condition in 
which it was when it began to move. Therefore, if nothing 
interferes with it, it will repeat the operation just described and 
continue to do so until its energy is expended. 

293. Eelative Positions of the Particles. Several important 
things are apparent from this discussion. In the first place, we 
note that when particle 1 has executed a complete vibration, the 
other particles along the wave have not yet done so. Each has 
performed only part of one vibration. Each successive particle 
has executed a smaller portion of one vibration than has the par- 
ticle ahead of it and a larger portion than has the one behind 
it. Thus, when particle 1 has completed its first vibration, 17 
is just ready to begin 
moving; 13 has executed 
J of a vibration; 9, J a 
vibration; 5, J of a vi- 
bration, and the inter- 
mediate particles inter- 
mediate fractions of one 
vibration. It is conven- 
ient to have a simple word for expressing this relation. There- 
fore, the particles are said to be in different phases of vibra- 
tion. The phase thus means the portion of one complete 

Fig. 178. 1 has Executed One ^Vhole Vibration 

WAVE MC/riON 305 

viMation that any particle has executed. So we may say that par- 
ticle 16 has a phase zero, 12 a phase of J vibration, 8 a phase 
of i vibration, 4 a phase of J vibration, and 1 a phase of 
1 vibration. 

294. Wave Length. Using this term "phase," we may make 
a general convenient definition of wave length; for we may say 
that a wave length is the shortest distance between any two par- 
ticles that are in the same phase. Thus the distance 1 to 17, 2 
to 18, and so on, is a wave length. 

295. Time of Vibration. It is often convenient to consider 
the time it takes a particle to execute one vibration instead of 
considering the numbers of vibrations per second themselves. When 
a vibratory motion continues to be repeated in equal time intervals, it 
is called periodic, and the time taken . by any particle in execut- 
ing one vibration is called the period of that vibration. Hence, when 
we are considering vibrations from this point of view, we may 
speak of a phase of a quarter period, of half a period, etc. 
Further, it is evident that if the source of vibration executes 10 
vibrations in a second, the time it takes to execute one vibra- 
tion, i.e., its period, is f ^ sec. Thus, in general, if the number 
of vibrations per second is represented by n, the period is always 

1 ^ * 

— sec. 


296. Velocity of Propagation. An important conclusion 
may now be drawn from the discussion of Fig. 178; for we 
note that while particle 1 has been executing one vibration, the 
disturbance has traveled a distance 1 to 17. This distance is a 
wave length. Hence, it is manifest that the disturbance travels 
along the row just the distance of one wave length while particle 
1 executes one vibration. If particle 1 executes n vibrations in 
a second, how far will the disturbance travel in that second? 
Evidently n wave lengths. But the distance traveled in one 
second measures the velocity (c/. Art. 2). Hence, we may express 
this result by saying that the velocity with which a wave travels is 


numerically equal to the prod/uct of the number of vibrations per 
second and the wave length. If v represents the velocity, n the 
number of vibrations, and / the wave length, then 

V = nl (14) 

This simple relation enables us to determine the velocity of 
the waves when we Can measure the wave length and the num- 
ber of vibrations per second of the source. This equation, how- 
ever, does not tell us how this velocity depends on the properties 
of the medium through which the waves travel. We can get a 
general idea of how the properties of the medium affect the velocity 
with the help of equation (4), Art. 27. For / = ma, therefore 

a = — . In this case / is the elastic force acting between two 

adjacent particles of the medium, m the mass of a particle, and a 
the acceleration given to the particle by the force /. Since a is 
proportional to /, the equation shows that if / is increased, 
particle 2 will have a greater acceleration; so it will fol- 
low faster after 1. For the same reason 3 will follow faster 
after 2, and so on ; therefore the disturbance must travel faster 
along the row of particles. 

On the other hand if the density of the elastic medium is 
greater, each particle will have a greater mass m. In this case 
the equation , shows that the particles will have smaller, accelera- 
tions, so each one will move more slowly and lag more behind the 
one just ahead of it; therefore the disturbance will travel more 

The exact relation of the velocity of a wave in a medium to 
these two factors, elasticity and density, has been determined 
mathematically and by experiment; and it has been found that 
the velocity v of waves traveling in a medium having an 
elasticity e and a density d is 




297. The Types of Waves. In the discussion thus far we 
have confined our attention to waves in which the motion of the 


particles is perpendicular to the direction in which the waves 
travel. When this is the case, the waves are said to be trans- 
verse. We have also pictured the tnotion of each particle as 
taking place along a straight line. These restrictions were in- 
troduced in order to simplify the discussion, though neither one is 
essential. Thus we may just as well have waves in which the 
paths of the particles are circles or ellipses in planes perpendicular 
to the direction of propagation of the wave; or we may conceive 
that the particles move back and forth in the direction of propa- 
gation of the waves. In this latter case the particles are alter- 
nately crowded together and separated, so that we have to deal 
with condensations and rarefactions of the medium instead of 
with crests and troughs. Waves of this type are called longi- 
tudinal. We shall leam more concerning both types of waves 
in the following chapters. 

298. Waves of Simple Shape. In this discussion it has been 
stated that we may draw conclusions as to the nature of the vibra- 
tion by a study of the shape of the wave. In Fig. 174 we have 
drawn a wave of particular shape. W^hat sort of vibrations werie 
executed by the body from w^hich these waves proceeded? From 
the simplicity of the shape of the curve we may imagine that the 
vibration must be of a simple type. Now, this type of vibration 
is that executed by a pendulum, as may be easily shown, by fasten- 
ing a small pencil to the bob of the pendulum and drawing a card 
under it in a horizontal direction, and in such a way that 
the pencil writes a trace on the card while the pendulum 
is vibrating. If we do this, we find that the shape of the curve 
obtained is the same as that shown in the figure. The pendulum 
has thus been made to construct a graph which represents the 
relation between the displacements and the corresponding times 
for its own motions. Vibrations of this type are called simple 
HARMONIC vibrations, and the curves that represent them 
graphically are called sine curves. 

Since every particle in the wave executes the same kind of 
vibrations as the source does, the shape of the waves that originate 
from a simple harmonic vibration will be that of a sine curve. 



FiQ. 179. 

Same Period, Amplitude, 
AND Phase 

When we have to deal with a simple harmonic motion of one 
definite period only, the corresponding waves are called homo- 
geneous. All other forms of waves are complex. It is perhaps 
unnecessary to remark that the ones with which we are actually 
familiar are in every case complex. 

299. Complex Waves. If all waves which we know in nature 
are complex, why do we study simple hompgeneous waves at all? 

In answer to this question, let 
us consider what happens when 
we h^ve two or more simple 
homogeneous waves traveling 
through the same medium at 
the same time. The result is most easily obtained from a dia- 
gram. Let us begin with the case shown in Fig. 179, and 
suppose that the two simple waves have the same period, am- 
plitude, and phase. The disturbance that results when these 
two waves are traveling through the same medium at the 
same time is obtained by adding as vectors the displacements of 
the particles. The result is shown by the lower curve in the 
figure. We note that the resultant is a wave of the same period 
and phase, but with twice the amplitude. 

Repeating this operation for two waves that have the same 
period and amplitude, but differ in phase by one-half period, we 
get the result shown in Fig. 180. This result shows that two such 
waves may be traveling in the 

same medium without giving 
any external sign of their pres- 

If we were to add together 

Fig. 180. Same Period and Amplitude, 
Opposite Phases 

two waves of the same period, but differing in phase by J of a 
period, or by any other fraction of a period, we should find that 
in every case the resultant w^ave has the same form as the con- 
stituent waves. Hence we may conclude that the addition of any 
number of simple homogeneous waves of a given period always 
gives a resultant which is also a simple homogeneous wave of the 
same period. 



Fig. 181. Addition of Waves of Different 

300. Wav«s of Different Shapes. If the addition of simple waves 
of the same period always gives as a result a simple wave, how may 
we produce waves of 


complex fonn? Let 
us see what the effect 
will be if we add to- 
gether two waves that 
have different periods. 
Take, for example, the 
two waves drawn in 
Fig. 181, one of which 
is twice as long as the 
other and has twice 
the amplitude. We 
note that the resultant 
R obtained by adding them together is a wave differing entirely 
in shape from the two component waves. Let us now add to 
this resultant a third simple wave, with period and amplitude each 
one-third of the first period and amplitude. The result is shown 
at R\ The resultant obtained by adding together waves 
whose periods and amplitudes have the ratios 1, J, ^ is shown 
in Fig. 182. The meanings of the curves will be perfectly clear 
on careful inspection. 

A study of these curves will make it apparent that we 
can produce waves differing greatly from one another in 
shape by adding together simple homogeneous waves which 

differ from one an- 
other in period as 
well as in amplitude 
and phase. That the 
number of shapes 
which may be pro- 
duced by the addition 
of such waves must 
be indefinitely great, 
may be realized by any one who considers that we have at our 
disposal, first, an indefinite number of possible periods; second. 

Fio. 182. Another Complex Wave 


an indefinite number of possible amplitudes; and, third, an in- 
definite number of possible phases. 

But even if the number of shapes that can be so artificially 
built up is practically without limit, is the converse proposition 
true, viz., that every shape that actually occurs in nature can be 
resolved into a series of simple homogeneous waves differing from 
one another in period, amplitude, and phase? This problem 
occupied the attention of mathematicians and physicists for a 
century and a half before its final solution was reached. That 
solution proved conclusively that this converse proposition is true, 
and shows us how to proceed in order to separate the compound 
wave into its component simple homogeneous waves. Hence, the 
importance of studying the nature of the simple waves becomes 
manifest, for it follows that all waves with which we are familiar in 
nature are built up of these simple homogeneous waves. The study 
of SIMPLE HARMONIC MOTION, which gives risc to these simple 
homogeneous waves will be deferred to the next chapter. 

301. Stationary Waves. There is still another kind of waves 
which we have not yet discussed, but which, nevertheless, 
merits attention. A jumping rope, one end of which is fastened 
to a tree, must be turned at a certain rate in order to swing properly. 
It is a matter of common observation that by turning it faster the 
rope may be made to break into two equal loops separated by a 
point where the rope moves very little. If the hand is turned 
still faster and the rope is long enough, the rope may be made to 
vibrate in three, four, or more parts. How are these loops 
formed, and why does the rope stay nearly still in certain places? 

An analysis of the operation will give us the answer. When 
the hand is moved periodically it impresses a certiiin vibration 
on the rope. The rope may be looked upon as a row of particles 
held together by elastic forces, and so the vibratory motion of 
the hand is propagated along the rope in the form of a wave, until 
it reaches the other end. What happens then? Does the wave 
give up all its energy to the tree, or is part of that energy reflected 
so as to travel back along the rope? You may easily show, by 
holding the rope rather tight and hitting it surldenly, that the 


wave is reflected ; for the hump raised on the rope by the stroke 
may be seen to travel to the far end of the rope and then to turn 
around and come back. Therefore, when you send a series of 
impulses along the rope they travel in the form of waves, are 
reflected at the further end, and return. We see, then, that if the 
series of impulses be continued, we shall soon have generated two 
trains of waves, the direct and the reflected, traveling along the 
rope in opposite directions. We may infer that, when this is the 
case, the result is the peculiar vibration that we get on the rope. 
Let us see if this is so. 

In the case we are considering, the two trains of waves have 
the same period and nearly the same amplitude, and are trav- 
eling along the rope at the same rate but in opposite directions, 
^hey are represented by curves A and B, Fig. 183, A moving to 
the right and B to the left. When the waves are in the positions 
indicated at V and D in the diagram, the resultant obtained by 
adding their displacements is shown by the thick black line. 

If now we conceive each of the two trains of waves to have 
advanced J wave length, A to the right and B to the left, their 
respective positions are those shown at W and E in the diagram; 
and the resultant will be the black line WE. 

Two more advances, of J wave length each, bring the two into 
the position shown at XyF and Y,G respectively. Clearly the re- 
sultants will be as there shown. A final advance of J wave 
brings the two into a position similar to their original positions, so 
that the resultant for Z^H is the same as that for V,D. 

W^hen these five resultants are superposed, we obtain the curves 
shown at RyNyC. The similarity between this figure and the rope 
in the case under consideration will at once be noted, for certain of 
the points never leave their positions of equilibrium, while others 
execute vibrations of greater amplitude. The positions of no 
amplitude are called nodes, while those of great amplitude are 
called LOOPS. Since the nodes remain at rest -with respect both 
to vibratory motion and also to the motion of propagation, such 
waves are called stationary waves. 

When we analyze the motion of the particles in these waves 
we see (1) that all the particles that move are in their positions 



of greatest amplitude at the same instant; and (2) that they are 
all in their positions of equilibrium at the same instant as shown 
by the straight line RNC. We note further (3) that all the par- 
ticles in one loop are in the same phase at the same time, but that 
their respective amplitudes are different, for the particles at the 

Fig. 183. Formation of Stationary Waves 

middle of the loop have a large amplitude, while those near the 
nodes have a small amplitude. Another fact worthy of remark 
is (4) that the particles in one loop have a phase that is differ- 
ent by half a period from that of the particles in either of the 
adjacent loops. And finally, we see (5) that the distance be-- 
tween two nodes is half a wave length. Thus it becomes 
Qlear that sii^h stationary waves are actually produced by two 


trains of waves of the same period traveling along the same row 
of particles in opposite directions. The relations between the pe- 
riod of vibration and the length of the loop will be taken up in 
a later chapter, for these stationary waves play a conspicuous part 
in the pheno^nena of sound, and so we shall have to discuss them 
further when considering that subject. 


1. Waves originate at a vibrating body. 

2. Waves bring us four kinds of infonnation: 1. As to the 
direction of the source. 2. As to the period of the vibrations. 
3. As to their amplitude. 4. As to their complexity. 

3. The characteristics of waves are direction of propagation, 
length, amplitude, and shape. 

4. A suitable medium is necessary for the transmission of 

5. The particles of a medium do not partake of the progres- 
sive motion of a wave, but merely vibrate about their positions of 

6. In progressive waves the successive particles are in different 
phases at the same time. 

7. Waves may be transverse or longitudinal. 

8. The velocity of propagation of waves is equal to the number 
of vibrations multiplied by the wave length, {v = nl.) 

9. The velocity of waves in an elastic medium is equal to the 

square root of the elasticity divided by the density, v = ^—, 

10. When waves are superposed, the resultant is the algebraic 
sum of the components. 

11. The simplest kind of wave is the simple homogeneous or 
sine wave. 

12. The vibrations that produce these simple waves are called 
simple harmonic vibrations. 

13. Waves of complex form result from superposition of 
simple homogeneous waves of different periods, amplitudes, and 


14. Waves of complex form may always be analyzed into a 
series of simple homogeneous waves. 

15. Stationary waves are produced when two trains of waves 
of equal period, but traveling in opposite directions, are super- 


1. Describe the motions of water at the point where a stone is 
dropped into it. 

2. What sort of information is derived frorii each of the four chief 
characteristics of waves? 

3. Describe the mechanism of wave propagation. Do the vibrat- 
ing particles partake of the progressive motion of the waves? j 

4. What two types of waves may we have? What is the distinctive 
feature of each? 

5. What can you say of the relative phases of the successive vibrat- 
ing particles along a progressive wave? 

6. How may wave length be defined with reference to phase? 

7. Does it seem reasonable to suppose that waves can travel 
through empty space? 

8. What is a simple homogeneous wave? From what kind of 
vibration does it originate? 

9. When two or more waves are traveling at the. same time in the 
same medium, how do we find the resultant wave? 

10. What sort of vibrations does a pendulum execute? How may 
we obtain a graph to show this? 

11. Explain how stationary waves are produced? 

12. What are nodes and loops? 

13. How is the distance between two adjacent nodes related to 
the wave length? 


1. What is the period of vibration of an oarsman who makes 40 
strokes per minute? Of a swhig that makes one complete oscillation 
in 4 sec? In 6 sec? Of a wagon seat that makes two complete vibra- 
tions in 1 sec? 

2. How many complete vibrations per sec does a tuning fork make, 
if its period is 2 Jo sec? What is the period of the waves started by a 
paddle-wheel that has 6 paddles, and makes 20 revolutions per minute? 
If these waves are 2 ft. long, what is tlioir velocity? 

3. Air and hydrogen have the same elasticity, under given eon- 
ditons, but air is 14.5 times as dense as hydrogen. Supposing sound 
to be a wave motion, ought it to travel faster or slower in liydrogen 
than in air? How many times? 


4. What is the length of the waves that are traversing a jumping 
rope 30 ft. long when it is vibrating in 1 loop? In 2? In 3? In 4? 
In 5? If the period is 1 sec when it is vibrating in 1 loop, what is the 
period in each of the cases just supposed? What is the velocity of 
each of the waves? 

5. Plot the following cases of simple homogeneous waves travel- 
ing in the same direction: 1. Two waves of equal period, amplitude, 
and phase. 2. Two waves of equal amplitude and period, but of 
opposite phai&e. 3. Two waves of equal amplitude and period, but 
with a difference of phase of J period. 

6. Plot the following cases of simple homogeneous waves travel- 
ing in the same direction: 1. Two waves of equal phase, one having 
half the amplitude and half the wave length of the other. 2. Combine 
this resultant with a third wave whose length and amplitude are each 
J that of the first wave. 

7. Were you ever out in a boat when the waves were running fairly 
high, or have you ever floated on your back among them? In that 
case you had an excellent opportunity to observe carefully the kind of 
motion that the water particles in the wave have. Did the water 
move you up and down only, or was there compounded with this up 
and down vibration another that was nearly horizontal? What was the 
resultant path or orbit of the water particles which were carrying you 
with them as they oscillated? 

8. Suggest a way for producing transverse waves in a long rubber 
tube. How may you produce longitudinal waves in it? Can water 
react elastically to forces that tend to compress it longitudinally as well 
as to forces that tend to displace it laterally? Do air particles cling 
together as water particles do? Can air resist both transverse and 
longitudinal stresses? If not both, which? 

9. In which direction (longitudinal or transverse) does a rubber 
tube offer the greatest elastic resistance, to a force producing a given 
displacement? Which kind of waves then (longitudinal or transverse) 
will travel faster in the rubber? Answer the same questions for 
Water, wood, brass. 


1. Throw stones of different sizes into a pool of water, and note the 
differences in the lengths and amplitudes of the corresponding waves. 

2. By moving both hands up and down with a regular period in a 
tub of water, see if you can produce stationary waves. Keep the 
hands a foot or £wo apart, and gradually change the period till the 
nodes and loops are seen at definite places in the water. 

3. With two companions and a kodak, go to a pond or lake and 


make a similar experiment. Instead of your hands use as sources of 
waves two long poles just alike, having nailed to them equal circular 
pieces of board. These will be easy to keep vibrating with equal 
periods. When two of you have practiced so that you can maintain 
the stationary waves, let the photographer of the party take a snap 
shot of the waves. By shortening one of the poles and reducing the 
size of the circular board on it, see if you can succeed in getting 
waves of forms that are more complex, but nevertheless definite. 

4. For many beautiful and simple home experiments on wave 
motion, read Prof. A. M. Mayer's Sound (Appleton, New York). By 
all means read Prof. J. H. Fleming's Waves and Ripples (E. & J. B. 
Young & Co., New York). 

This is a series of experimental lectures delivered to young people 
at the Royal Institution, London (where Davy, Faraday, and Tyndall 
worked). It is a most fascinating account of waves in water, air, and 

5. If you are in the manual training class or have a shop of your 
own, get together with some of your classmates and make for the 
school a Kelvin wave model as described in Michelson's Ldght Waves 
and their Uses (University of Chicago Press) pp. 5 and 6. Read also 
pp. 1-13. 

6. Another excellent wave motion model which you can easily 
make is described in Jones's Heat, Lights and Sounds p. 238. Read 
also pp. 236-241. 

7. Let a companion hold one end of a clothes line while you hold 
the other, and with it produce the result predicted theoretically by 
the diagrams in Fig. 183. Let another companion who is a photog- 
rapher try to get some snap shots of the line while vibrating in 1, 2, 
3, 4, or more loops, and make lantern slides of them for the school 



Note. — The authors recommend that this chapter be used only for 
informal discussion on the first reading. If time is short it may be 
omitted altogether. 

302. Relation to Uniform Circniar Motion. The study of 
simple harmonic motion is made much easier by first consider- 
ing the relation that exists between this type of motion and 
uniform motion in a circle. In 
order to make this relation clear, 
let us conceive that we have a 
small body traveling with uniform 
velocity v, in the horizontal cir- 
cular path ABCD, Fig. 184. To 
one looking at this motion from 
above, the path of the body is 
seen to be circular and its velocity 
uniform. But if the motion be 
observed from a point in the 
plane of the circular path, and 
at a distance from the circle in 
the direction CA, the body will appear to be moving back and 
forth along the straight line BD, and its motion will no longer 
appear uniform. Thus, when the body is passing the points B 
and D in its circular path, it will appear to be at rest. On the 
other hand, when it is passing the points A and C, it will appear 
to be moving with a speed which is the same as its uniform 
speed V in the circular path. At intermediate points its speed 
will appear to vary between these two limits, i.e., between 
and V, 

When we observe the uniform motion of the particle in this 
latter way, it appears at each instant as if it were projected on the 


Fig. 184. Simple Harmonic Motion 


straight line BD, ix., as if it were at the foot of a perpendimdar 
dravm from it to BD, So we can readily understand how the 
same effect would be produced on the distant observer if we 
replace the body moving uniformly in the circular path by 
an equal body that moves back and forth along the line ED 
in such a way that the position of this second body at any 
instant is the projection on the line BD of the position of the 
first body at the same instant. Let us conceive this to be done. 
It then remains for us to consider what forces must be applied 
to the body moving in this way along BD, in order to produce the 
required motion. The problem is not. so diflScult as at first sight 
it may appear. 

303. Force and Displacement. Suppose the first body is at 

any point E (Fig. 184) of its circular path. The second body 

must then be at F, the projection of E on BD, What force is 

acting on E at this time? We have learned in Chapter V that the 

force is directed toward the center of the circle, i.e., along EO, 

and is numerically equal to , in which m is the mass of the 

body, V the uniform velocity in the circumference, and r the radius 
of the circle. Let the magnitude and direction of this force be 
represented by the vector EM. We may now conceive this 
force to be resolved into two components, one in the direction 
EF, perpendicular to BD, and the other in the direction 
EH, parallel to BD, These components will then be rep- 
resented by EP and EN, It is clear that the component 
EP has no effect on the motion of the body at E in the 
direction BD, It is also clear that if we allow a force equal 
to the component EN to act on an equal mass at F, the motion 
produced along BD will be the same as the motion in the direc- 
tion BD of the body at E, Hence, the force that must be applied 
to the second body at F, in order that it may always be at the 
projection of E on BD, is represented by this component EN, 
But what is the value of this component? From the similar 

triangles, EMN and EOH, g = gor. EiVT = ^^ >< |g- 


But EO is the radius v of the circle and EH = FO is the distance 
of the second body from 0. If this distance be denoted by d, 

EN = EM X — , but EM = — , therefore substituting this value 

TfVu d 

for EM we have EN = — X -. But EA'' represents the force • 

r r ^ 

that must be applied to the body at F in order to make it move 
in the required manner. If we denote this force by /, we have 

In a given case m, v, and r are constant, therefore the force that 
must be applied at each instant to a body in order to make it 
vibrate in the required manner, is proportional to the distance of 
the body from the center of its swing. If we call this distance the 
DISPLACEMENT, and define the motion as simple harmonic 
MOTION, We reach the conclusion that when a body is vibrating 
in simple harmonic motion, the force at any instant is propor- 
tional to the corresponding displacement, 

304. The Sine Curve. Since the value of the ratio — depends 

on the size of the angle EOH, it is possible to express 
the force / in terms of this angle instead of the displacement. To 
do this, we name this ratio the sine of the angle EOH, i.e., we 
define the sine of an angle as the ratio in a right triangle of the 
side opposite the angle to the hypothenuse. It is for this reason, 
and also because the simple homogeneous waves considered in 
the last chapter are produced by this simple harmonic motion, 
that the graphs representing their shapes are called sine curves. 
The abscissas of the sine curve represent the angles and the ordi- 
nates represent the values of the corresponding sines, 

305. Illustrations. We can now realize why simple harmonic 
motion is of so great importance in science, for vibrations are 
usually produced by elastic forces, and these are proportional to 
the displacements. One of the simplest cases of such vibration is 
that of a weight on the end of a spiral spring, Fig. 185. In this 



case we may easily prove that the force is proportional to the 
displacement; for when we hang a weight of 100 gm on the end 
of the spring and measure the displacement produced, and then 
repeat the operation with a weight of 200 
gm, we find that the displacement in the 
second case is twice what it is in the first, 
and so on. 

306. Period. One other point remains 
for consideration: How does the time of 
vibration depend on the mass of the vibrat- 
ing body and the force acting to bring it 
back to its position of equilibrium? We 
may find the answer to this question as 
follows: When body 2 which moves along 
the diameter BD, Fig. 184, is passing the 
point O of its swing, it is evident that its 
velocity is the same as that of body 1 at 
A ; i.e., the velocity is v. But if T repre- 
sent the timie it takes body 1 to travel 
once around its circular path, i.e., if T 
represent the period, this velocity will be equal to the circum- 

ference of the circle divided by T, i.e., v = "V^* ^^* since at 

the velocity of body 2 is also v, or —pp—, its kinetic energy at 

this point is found by substituting this value in equation (6), Art. 
39, i. e.; 

^^~2 ~^\Y) ~~2r'' 

When, however, body 2 reaches the point B, its velocity is 
reduced to 0. Hence, at B it has no kinetic energy, but this 
energy has been converted into potential energy. This potential 
energy, as we learned in Chapter II, is equal to the work done in 
bringing the body to the position B, Now, this work is the force 
/ X the displacement. So if we let F represent the force acting on 
body 2 to bring it back to O when its displacement is 1 cm, the 

Fig. 185. Force is Pro- 
portional TO Dis 



force / acting at B to cause its return will be Fr, because it is 
proportional to the displacement OB, and OB is r cm from 0. 
Now, this force increases from the value to Fr as the displace- 
ment increase^ from to r. Therefore, in order to get the work 
done in moving the body from to 5, we may assume that this 
variable force is replaced by a constant one. The numerical 
value of the constant force that will do the same amount of work 
in this case is the average of the forces at and at B; i.e., this 

+ Fr Fr 
force is equal to — - — = — . Therefore, the work done in mov- 

ing the body from to J5 is this force — multiplied by the dis- 

placement r, i.e., W = -^. This work is equal in value to the 

potential energy of the body at B; and this potential energy 
is, as just stated, equal to the kinetic energy at 0. Therefore, 

4w^mr* Ff^ r^ . . .,. .. « ,r» ^ ^9^ rm 

—K?p2 = ~^' Solvmg this equation for !P, we have Air — = r*, 

and finally /~~" 

T = 2.^|, (16) 

i.e., the time taken in executing one complete vihration is equal to 
2ir multiplied by the square root of the quotient obtained by di- 
viding the mass of the body by the force necessary to displace it 
1 cm from its position of equilibrium. This force per cm is 
called the force constant of the system. 

It is easy to see that the expression on the right-hand side of 
the equation represents a time, for its symbol in tenns of gm, cm, 
and sec evidently is the square root of gm divided by that for 
dynes per cm, or 


g = V sec^ = seCk 
gm cm 

sec cm 

307. Pendulum. There are several other important cases in 
which this relation can be applied, besides that of a spiral spring. 
Most important among these is that of the pendulum. We will now 
take up the consideration of this case. Suppose our pendulum 



consists of a ball of lead A, of mass m, suspended on the end of a. 
wire, Fig. 186. Let the distance between the point of suspension 
and the center of gravity of the ball be denoted by I. The 
mass in this case is clearly m, but what is the force constant? 

As is well known, the pendulum 
when set in motion moves back- 
ward and forward along the arc 
ABC, while its motion, to be 
strictly simple harmonic, must 
be along the straight path AC. 
But if the arc ABC is small 
compared with the length Z, the 
difference between this arc and 
its chord AC becomes so small 
that we may, without apprecia- 
ble error, consider that the actual 
path ABC is equal to the chord 
AC, and that the motion is har- 
monic. The displacement of the 
mass is, then, the distance AD 
in the figure. The question 
therefore is. How great a force 
is acting on the mass m when this 
N displacement equals 1 cm? To 
Fig. 186. The Pendulum Diagram answer this question, conceive 

the pendulum swung to the posi- 
tion A, in which its displacement equals 1 cm. The only force 
involved is the weight of the ball acting vertically downward. Let 
the vector AM represent this force. We now conceive this force 
to be resolved into two components, AN and AP, one in the 
direction of the wire OA and the other perpendicular to it. The 
component AN merely causes a tension in the wire, while the 
other component AP produces the motion along the arc AB. 
What, then, is the value of the component ^P? 

Since the triangles AMP and ADD are similar (Why?) we 


A^ ^AD 
AM ~ OA' 

therefore, AP = 


but AM repre- 


sents the weight of the body, i.e., mg, AD = I cm, and OA 

= /, therefore AP = -^. This is the force corresponding to 

unit displacement, therefore it is the force constant for this sys- 
tem. When we have substituted this expression for the force 
constant F in our equation (16), the result is 

mq \ n 

We thus reach the conclusion that the time a pendulum takes 
to execute one complete vibration is numerically equxil to 2ir mul- 
tiplied by the square root of the quotient obtained by dividing the 
length of the pendulum by the acceleration of gravity. It must 
not be forgotten, however, that this is strictly true only when the 
displacement is so small that the chord ADC and the arc ABC 
are sensibly equal and when the, mass of the wire is inconsid- 

308. Uses of the Pendulum. This is one of the most important 
relations in physics, for it furnishes us with a simple and very 
accurate method of determining g, the acceleration of gravity. 
Thus, if we solve this equation for g, we get 

9 ~ rp2 f 

and since we can easily measure the length of a pendulum and 
ahso its time of vibration, we obtain the value of g immediately. 

This equation is also of use in proving with great accuracy 
that g is the same for all bodies at a given place; for if we make 
a series of pendulums all of the same length, but whose bobs are 
made of different substances, and if we find that they all 
vibrate in the- same time, we must conclude that at a given place 
g is the same for all masses. This experiifient was performed by 
Newton, and later with greatest accuracy by Bessell, and the 
results show that all pendulums of the same length, no matter 
of what substance they are made, vibrate at a given place in the 



same time. Therefore we are justified in comparing masses by 
comparing their weights, as stated in Chapter II. 

Furthermore, since all pendulums of the same length vibrate 
at a given place in the same time, the pendulum furnishes a most 
ccmvcnient method of measuring time. The enormous importance 

of the pendulum to 

mankind in this respect 
is so familiar, that we 
need do no more than 
call attention to it. 

In the case of the 
pendulum, it is custom- 
ary to call its period 
the time taken in mov- 
ing from one end of its 
swing to the other, not 
the time taken to com- 
plete a whole vibra- 
tion. But since this 
time is half that re- 
quired for a complete 
swing, the equation for 
the pendulum is usually 
written : 



Fig. 187. The Proof that the Earth Rotates 

309. The Foucault 
Pendulum. The 'pen- 
dulum also furnishes 
The experiment that 
He sus- 

o means of proving that the earth rotates. 

shows this was first performed by Foucault in 1851 

pended a ball of lead, having a mass of 28 Kg, on a steel wire 

67 m long in the dome of the Pantheon in Paris, Fig. 187. On 

starting the pendulum into vibration, it was found that the plane 

in which it swung turned with reference to the building, and the 

amount of this turning could be measured on the large circle 


beneath the bob, for the pendulum at each vibration would knock 
down parts of a ring of sand which had been piled up around the 

The explanation of this phenomenon is as follows. On 
account of its inertia the pendulum swings in a plane that has a 
fixed direction in space; and, therefore, as the earth turns with 
reference to this fixed plaije, this plane appears to turn with refer- 
ence to the earth. If such a pendulum were suspended directly 
over the north or the south pole of the earth, its plane of vibration 
would turn once around in 24 hours. On the equator its plane 
would not turn at all, and in intermediate latitudes it would 
oscillate back and forth every day through an angle that depends 
on the latitude. 


1. When a body moves with simple harmonic motion, the 
force that acts to return it to its position of equilibrium is 
proportional to the displacement of the body from that position. 

2. Elastic forces are proportional to the displacement. 

3. A body vibrating under the action of elastic forces executes 
simple harmonic motion. 

4. The periodic time of a body executing simple harmonic 
motion is equal to 2 w times the square root of the mass divided 
by the force constant. 

5. A pendulum when its displacement is small vibrates in 
simple harmonic motion. 

6. The periodic time of a pendulum for a single swing is 

equal to tt multiplied by the square root of — . 

7. The pendulum furnishes the most accurate method of 
determining g, 

8. With the pendulum we may prove that g is the same for 
ail masses at a given place. 

9. The pendulum is our best measurer of time. 

10. The pendulum furnishes us with a means of proving that 
the earth is rotating. 



1. What relation exists between forces and displacements when a 
body moves with simple harmonic motion? How is this proved? 

2. Do bodies vibrating* under the action of elastic forces execute 
simple harmonic motion? Why? 

3. How does the periodic time of a body vibrating with simple 
harmonic motion depend on the mass of the body? 

4. What is the force constant? How does the periodic time depend 
on it? 

5. Upon what two factors does the periodic time of a pendulum 

6. How may the pendulum be used: 1, to measure g; 2, to prove 
that we may compare masses by comparing weights; 3, to measure 
time; 4, to show that the earth rotates? 


1. Draw a circle 6 cm in diameter: Beginning at the point cor- 
responding to A, Fig. 184, divide the circumference into 12 equal parts, 
and draw perpendiculars from the end of each of these arcs to the diam- 
eter A C. Plot a graph in which the abscissas represent the lengths 
of the arcs, measured from the point A, and the ordinates are the 
lengths of the corresponding perpendiculars. Does the curve obtained 
resemble that of Fig. 174? Repeat the construction, using the same 
scales, but with circles 3 cm and 2 cm in diameter, taking care to have 
the origin of coordinates for all the curves fall on the same vertical 
line. Graphically add the three curves together. Does the resultant 
resemble the curve R' in Fig. 181? 

2. Draw a circle of 5 cm radius. From the point corresponding to 
A, Fig. 184, lay off arcs of 15°, 30°, 45°, 60°, 75°; and drop a perpen- 
dicular from each point thus determined to the diameter A C. Measure 
the lengths of these perpendiculars in cm, divide the numbers that 
represent those lengths by 5, and compare the numbers thus obtained 
with those given in a table of natural sines opposite the same degree 
number, i e., 15, 30, etc. May the sine of an angle be defined as the 
lengtli of such a perpendicular in a circle whose radius is unity? Com- 
pare Arts. 298 and 304 and problem 1 and see if you can understand 
the meaning of the term sine curve. 

3. What is the length in cm of a pendulum that beats seconds at sea 
level in New York? How long must ^ pendulum be in order to vibrate 
in 2 sec? In 3 sec? Write an equation expressing the relation 
between the times t and t' of two pendulums and their lengths I and Z'. 

4. The spring Fig. 185 is elongated 5 cm by a 100 gm weight. 
What is its force constant? If amass of 250 gm is suspended on the 
spring and set vibrating up and down, wliat will be its period? 


5. A circular brass disc is rigidly fastened to a stiff steel wire pass- 
ing through its center and perpendicular to its surface. The wire is 
held vertically and its upper end firmly clamped so that the disc hangs 
in a horizontal plane. If the disc be turned through a small angle 
^bout the wire as an axis, and then released, it will execute rotary 
or torsional vibrations. With the help of Art. 86, tell what will in 
this case correspond to the force constant. What must take the place 
of the mass in the equation of vibratory motion in Art. 306? What, 
then, is the equation for determining the period of vibration? 

6. A moment of force whose numerical value is 237 X 10''' is required 
to twist the disc of problem 5 through an angle of 1 radian. If the 
moment of inertia of the disc has the value of 6 X 10^, what is the 
time of vibration? 

7. A disc suspended as in problem 5. has a diameter of 20 cm; a 
force of 10® dynes, acting tangentially at each end of a diameter, is 
required to give it an angular displacement of 1 radian. If its time 
of vibration is 1.5 sec, what is the value of its moment of inertia? 

8. A bar magnet mounted on a pivot, like a compass needle, and 
deflected through an angle of 1 radian from the magnetic meridian, 
tends with a moment of force whose value is 990 to return to that 
meridian. If the moment of inertia of the magnet, has the value 400, 
what will be the period of oscillation of the magnet when it is released? 

9. The moment of force of the bar magnet in problem 8 depends 
on the mutual action between the magnetic fields of the magnet and of 
the earth. If the strength of the magnet's field remains constant, 
how will the period of oscillation of the magnet be changed if the 
strength of the earth's field is doubled? 


1. See how nearly you can determine your own mass by swinging 
in a swing, determining with your watch your period of vibration, 
and getting a friend to measure with a spring Ibalance the number of 
dynes necessary to pull you in the swing a measured number of centi- 
meters from the position of equilibrium. If the spring balance is 
graduated in pounds, remember that 1 pound-force =445,000 dynes. 

2. Tie both ends of a rope to the branch of a tree about 10 ft. from 
the ground, so that the rope hangs in a V whose point is about 5 ft. 
from the ground. From the point of the V suspend by a single cord 
a tin can, so that it almost touches the ground. Punch a small hole 
in the bottom of the can, fill it with water, feet it to swinging, and see 
what sort of curves the water will draw on the ground. Similar 
experiments are described in Mayer, On Sound (Appleton, New York). 


810. Sources of Sounds. Of all the phenomena of nature 
none, perhaps, is better known or more universally recognized 
than the fact that sound always originates at some vibrating body. 
Even an infant knows that he must shake his rattle to make it 
sound, and the vibrations of a bell or drum, when they are sound- 
ing, are easily felt. 

Other familiar facts concerning sound are the following: 
1. We can in some way tell in what direction the source of sound 
lies. 2. We recognize differences in the pitch of sounds, some 
seeming high and shrill, others low and deep. 3. We recognize 
differences in the intensity of sounds, some being loud and strong, 
others soft and weak. 4. We recognize differences in the quali- 
ties of sounds, i.e., we are able to distinguish at once between the 
tone of a violin and that of a piano, and can even recognize one 
another by our voices in the dark or over a telephone. 

311. Sound a Wave Motion. With these facts clearly before 
us, many interesting questions arise. How does sound get from 
the vibrating body to us? How do the vibrations of sounds of 
different pitch differ? What governs the intensity of sound? 
What characteristics of the vibrations of the source correspond 
to the differences in tone quality? Let us proceed to find the 
answers to these questions. 

The first conclusion that we draw from the facts just men- 
tioned is that the information which sound brings us concerning a 
sounding body might be gained from a wave motion. Hence it 
seems plausible to assume as a hypothesis, that sound travels from 
its source to us by a wave motion. But if it is a wave motion,what 
is the medium in which it travels? If we place an alarm clock 
under the receiver of an air pump, we find that the alarm is no 


SOUND 329 

longer audible when the air is pumped out. This experiment 
proves that ordinarily air is necessary for the propagation of 
sound, and if we conclude that sound is a wave motion in air, 
it lends added weight to the conclusion that waves can not be 
propagated unless a suitable medium is present to transmit them. 
Other facts- point to the conclusion that sound is a wave motion 
in the air; for the presence of these waves in the air may be detected 
by suitable apparatus, such as membranes and sensitive flames. 
Another proof of this fact is derived from the velocity with which 
sound travels, for this velocity can be measured by firing a gun at 
one place and noting at another, distant place the time that elapses 
between seeing the flash and hearing the report. It can also 
be calculated from the properties of air, for in Chapter XIV we 
have learned that the velocity of waves in an elastic medium is 

equal to -^ -^ and both e and d for air are capable of measure- 
ment. . If the two values obtained from these two different meth- 
ods agree, we are well justified in saying that sound is a wave 
motion in air. 

312. Sound Waves are Longitudinal. It will be interesting 
to calculate the velocity of sound waves in air from the formula. 
In order to do this, we must determine what we mean by elasticity, 
and this necessitates our knowing what kind of wave motion 
sound is. Now, in Chapter XIV we have found that waves may 
be either transverse or longitudinal. We there learned that when 
a medium transmits transverse waves, the forces brought into 
play are those that resist a sideways displacement. Hence, since 
air presents no elastic force that resists a sideways displacement, 
we conclude that it can not transmit transverse waves; yet since 
it offers a large elastic resistance to compression, it can transmit 
longitudinal waves with a large velocity. Therefore we conclude 
that the sound waves are probably longitudinal, and will proceed 
on this assumption to find what their velocity is. 

813. The Velocity of Sound. In order to calculate the ve- 

locity of sound with the help of the equation v =^ \ T> ^^ must 



first consider how the elasticity of the air is measured. The elas- 
ticity of any substance may be defined as the ratio of the pressure 
that produces the change, to the change per cm' produced. In 
the case of air, the change produced by applying pressure is a 

, . , ii # 1. . pressure applied 

change m volume: therefore, for air, e = —, — ^-f \ 

° " change per cm'' m volume. 

The numerical values of these quantities may be found experi- 
mentally by applying a measured pressure to air confined in a cylin- 
der and measuring the corresponding changes in volume. The 
results of such experiments show that, for rapid compressions 

like those of sound, a pressure of 14200 ^ is required to pro- 
duce a change of 0.01 ^g in volume. Hence, for air, 

. = li^= 142X10' ^^ 
O.Ol cm^ 

Since the density of air at 0° C and 76 cm atmospheric pres- 

1 1 42 y 10* 
sure is 0.001293, we have v ^ ^^ ^ = 33150. It is to 

be noted that the numerator is — — ^ and the denominator ^-^, 

cm* cm'* 

1 .1 P .1 .• , . gm cm ^ cm' cm* ^, ,^ 

and theretore the quotient is -^^ « X = — ?. i he result 

^ sec'' cm'' gm sec^ 

is thus- seen to have the symbol ^, as it should have if it is a 


The results of many experiments in which the velocity of sound 

has been measured by the method of firing a cannon and by 

other methods, show that this velocity is 33170 ^ under the 

conditions of temperature and atmospheric pressure specified. 

Since the calculated value agrees so well with the observed value, 

we may conclude that sound waves are longitudinal waves in 


314. Resonance. Another striking proof of the fact that sound 
is a wave motion in air may be given with a pair of tuning forks 
which are tuned so that they have exactly the same periods of 
vibration. If one of the forks is set into vibration, the other, 
though placed at some distiince from it, will begin to vibrate, so 

SOUND 331 

that it can be plainly heard if the first one is stopped. It must, 
therefore, have been set into vibration by the regular pulsations 
of the air that are started by the first fork. The little pushes of 
the successive waves are applied to it just at the proper time, so 
that their sum finally produces an appreciable motion of the 
second fork. Every child who has pushed a heavy person in a 
swing knows how the little pushes are able to set the swing into 
vibration if only they are properly timed. So with the two tuning 
forks; when the two forks have the same period of vibration, 
the little pushes of the air waves from the first fork reach the 
second fork at just the proper intervals, and thus set it into vi- 

When a body is thus set into vibration by waves of the same 
period as those which it is itself capable of sending out, its vibra- 
tions are said to be sympathetic, and the phenomenon is called 
RESONANCE. Every body, when vibrating freely, has a definite 
period of vibration peculiar to it. This period is called its nat- 
ural PERIOD. The period of the waves that act on a body to set 
it into vibration by resonance, is called the impressed period. 
The principle of resonance, then, is generally stated as follows: 
A body may he set into vibration by resonance when its natural ^period 
agrees with the impressed period, 

315. Effect of Temperature Changes. One further point 
remains for consideration, viz.: Is the velocity affected by a 
change in temperature? Evidently it is, since heating the air 
expands it and thus makes its density less, and a decrease in the 
value of the density rf means an increase in the value of the ve- 
locity V. Therefore, sound travels faster the warmer the air 
is. It is easy to show that the increase in velocity is 60 ^ for 
every rise of 1° C. in temperature. 

316. Noises and Musical Notes. Having thus proved that 
sound is a wave motion in air, let us pass on to a study of a 
vibrating body that produces sound. But before entering on 
this study, it will be well to make a distinction between noises and 
musical notes. For a noise is a confused jumble of sounds — 


an irregular and mixed phenomenon without definite period of 
vibration, while in the case of musical notes we have definite 
periods of vibration; and so the numerical relations are more uni- 
form, and lend themselves better to systematic investigation. 
Therefore, in what follows, we shall confine our attention solely 
to the musical notes, and whenever ths word ''sound'' is hereafter 
useij, a continued and regular sound of definite period, i.e., a 
musical note is meant. 

317. The Piano. With this limitation of the meaning of the 
word, we may safely say that a piano is one of the most familiar 
of all sources of sound. Let us then begin our investigation by 
noting some of the features of this instrument. On opening a 
piano, we find that there are inside it a number of steel wires 
of varying diameters, lengths, and tensions. If we strike a key, we 
observe that a small hammer flies up and strikes one or more of 
these wires. We further note that the wires struck are set into 
vibration, and that we hear the tone as long as this vibration 

Another fact that we notice is that the long and thick wires 
correspond to the lower keys on the keyboard, and emit, when 
vibrating, the tones of lower pitch in the musical scale. Such 
observations as these lead us to ask many questions. What 
relation exists between the lengths of the strings and the pitches 
of the corresponding notes? Why are there just eight notes in 
an octave? What is it in the sound that enables us to distinguish 
between the tones of a piano and those of a violin? Why do we 
call certain combinations of notes harmonious and others dis- 

318. Pitch. As has just been stated, we notice that the long 
strings in the piano are the ones that produce the tones of low 
pitch. We also observe that these * long strings vibrate more 
slowly than the shorter ones; i.e., they execute fewer vibrations 
per second. We therefore infer that pitch is in some way related 
to the number of vibrations per second. That this is really the 
case may easily be proved by mounting a toothed wheel on an 



Fig. 188. The Syren 

axis and revolving it. If we hold a card so that it strikes lightly 
upon the revolving teeth, we notice that the tones produced by 
the wheel are different for different speeds. Since each tooth 

causes a vibration when it strikes 

the card, we must conclude that 
the difference in the pitch pro- 
duced is due to the different 
number of vibrations when the 
speed of rotation changes. 

Another method of proving 
this same thing is this: Take 
a cardboard, or thin metal disk, 
and punch two or three rows 
of equidistant holes around its 
outer edge (Fig. 188). When we 
blow on one of these rows of 
holes while the disk is rotating 
rapidly, we notice that a tone is produced which is different when 
the number of holes in the rows is different. But since, when the 
disc is rotating uniformly, a difference in the number of holes 
means a difference in the number of pulses or vibrations forced 
on the air each second, it appears that pitch depends on the num- 
bers of vibrations per second. 

319. Musical Intervals. But we can prove more than this 
with these instruments; for we can show that definite simple 
relations exist between the vibration numbers of the notes on 
the piano. Thus, if the numbers of holes in two different rows 
on the rotating disc are related as 1 to 2, we perceive that the 
corresponding notes are one octave apart (from do to do); if the 
numbers of holes are related as 2 to 3, we find that the two notes 
are a fifth apart {do to sol). Similarly, if the notes given by the 
two rows of holes are do and fa, the corresponding numbers of 
holes in the rows are found to be related as 3 to 4. So it 
appears that the numbers of vibrations of the notes on the piano 
are related to one another by simple ratios. But before we take 
up the question as to the reasons for the existence of these simple 


numerical relations among the notes of the musical scale, we 
must stop to discuss briefly how the lengths and sizes of the strings 
affect the number of vibrations of the tones. 

320. The Laws of Strings. The laws of vibrating strings 
were discovered experimentally by Mersenne (1588-1648) in 
1644. These laws are merely statements of the numerical re- 
lations that appear in the well known facts that, other things 
being equal, the longer a string is, the more slowly it vibrates; 
the thicker and denser it is, the more slowly it vibrates; and the 
greater its tension, the faster it vibrates. They are: Otfier things 
being eqiml, the number of vibrations per second executed by a 
stretched string is: 

1, inversely jyroportional to its length; 

2, inversely proportional to its thickness; 

3, directly proportional to the square root of its tension; 

4, inversely proportional to the square root of its density. 

These laws are all illustrated by the strings of musical instru- 
ments. The short, thin, tightly stretched strings on the piano 
are the ones that give the high notes, while those for the low notes 
are longer, thicker, and not so tense. The same is also true of 
the violin, the cello, the banjo, and all other stringed instruments. 
In these latter instruments, the strings generally all have the 
same length, and on a given string the notes of higher pitch are 
produced by shortening the string by pressing the finger on it. 

321. Vibrating Rods. Similar relations are found to exist 
for the case of elastic rods fastened at both ends or supported 
in other ways. Rods may vibrate either transversely or longi- 
tudinally, and the vibration numbers are different in the two cases. 
This may be shown by clamping a metal or wooden rod about 1 
m long and 0.5 cm in diameter in the center and then setting it 
into vibration first transversely by striking it, and then longi- 
tudinally by rubbing it with a damp cloth. In the former case 
the vibrations will be slow enough to count; and other things 
being equal j the vibration numbers are inversely proportional to 
the squares of the lengths. In the latter a note of high pitch is 

SOUND 335 

produced and, other things being equal, the vibration numbers 
are found to be inversely as the lengths. In this case the vibra- 
tions are too rapid to be seen, but they may be shown by means 
of an elastic ball or button suspended so it will just touch the 
end of the rod. Thus it appears that the different ways in which 
rods may vibrate are many, since the number of vibrations de- 
pends not only on the dimensions of the rod, but also on the way 
in which it is supported and the manner in which it vibrates. 

Tuning Forks. One case of vibrating rods is of great 
practical importance, namely, the tuning fork. This instru- 
ment is universally used as a standard of pitch. Its vibra- 
tions are simple harmonic, as may readily be shown by fastening 
a light wire to one of the prongs and allowing the fork to trace 
its motion on a moving piece of smoked glass. The resulting 
curve will be found to resemble closely a sine curve (Fig. 174). 

Organ Pipes. One more class of vibrating bodies re- 
mains for, consideration, namely, organ pipes. In this case the 
vibrating body is a mass of air inside the pipe. This column of 
air may be regarded as a rod of air and its possible vibrations 
investigated, as in the case of rods. Here also the vibration 
number of the note given by such a column of air is inversely 
proportional to the length of the column. It varies also With the 
density of the air, but in all practical cases the changes in the 
density of the air, due to changes in atmospheric pressure, have 
so small an effect that they may be neglected. 

324. Air Columns as Eesonators. Since the air in an organ 
pipe has a definite mass and shape, it must, like all other bodies, 
have a natural period of vibration. Therefore, if we impress on 
this air a vibration whose period agrees with its natural period, 
the air will be set into vibration by resonance (cf. Art. 314). 
This resonance of an air column may be shown by holding a 
vibrating tuning fork of a certain pitch over the top of an open 
organ pipe of the same pitch. The air in the organ pipe is set 


into vibration by resonance, thus strengthening the tone given by 

the fork. The boxes on which tuning forks are usually mounted 

are made of such a size that the natural period of the air in them 

agrees with the period of the fork. When the 

y^] ] fork vibrates, the air vibrates by resonance, and 

thus the intensity of the tone is much increased. 

We may now form some idea as to how the organ 
pipe is made to operate. Air, under pressure, is 
admitted to the small air chamber a, Fig. 189, 
whence it is blown through the slot s in such a way 
as to strike the tongue L This stream of air blow- 
ing across the tongue produces a vibratory motion 
of the air, and when matters are so arranged that 
this vibratory motion has the same period as the 
najtural period of the air in the pipe, that air re- 
sponds by resonance. Toy whistles, flutes, and 
some other wind instruments work in a similar man- 
ner. The air in the tube of such an instrument 
always acts as a resonator and serves to strengthen 
the vibrations produced by blowing across an edge 
I I .'t of some sort. In the case of the clarinet and sax- 
1 *^^ ophone the vibrations are produced by a thin elastic 
i=7lVTr5 strip of metal called a reed, and in the case of the 
cornet and other horns, by the lips of the per- 
former, but the resonant air column is also essen- 

Organ pipes are made of definite length, and 
therefore each one has a definite pitch, so that we 
must have a separate pipe for each note. In the 
flute, however, the length of the air column may be 
varied by opening or closing holes in the tube with 
the fingers; so the flute can be made to produce 
various notes. In the trombone the length of the 
air column" is varied by a slide, which may be pushed out or 
in, thus lengthening or shortening the tube. Other wind in- 
struments will be found, on examination, to operate somewhat in 
a similar manner. 

Fig. 189. Open 
Organ Pipe 

SOUND 337 

325. Intensity. The pitch of a note has just been found to 
depend on the vibration number of the source of sound, and 
this vibration number is inversely proportional to the wave 
length in the air. Thus pitch and wave length are closely 
connected with each other. What in the waves corresponds to 
intensity of vibration in the source? The intensity of a wave 
must depend on the intensity of vibration of the source of the 
wave, and this latter is greater the greater the amplitude. Thus 
the amplitude of the sound waves tells us in a general way of the 
intensity of vibration ,of their source. It may readily be shown 
that the intensity is proportional to the square of the amplitude. 

The intensity of sound from a sounding body — a stretched 
string, for example — may be increased by changing the way in 
which the string is mounted. Thus, if the string is stretched 
between two heavy blocks of iron, the sound from it is not very 
intense, because the string has a small area, and so it slips, as it 
were, through the air without imparting much energy to it. But 
if the string is stretched over bridges on a large thin board, the 
bridges and the board are set into vibration by the string, and, 
since the board has a large area, a large amount of energy is 
transmitted to the air by it. Therefore the sound from the 
string is louder. Such a board is called a sounding board. 
The air in the tubes of organ pipes and other wind instruments 
serves a like purpose, as has just been mentioned, for the large 
mass of this air enables them, when set into vibration, to transfer 
more of their energy of vibration to the surrounding air, and so 
to increase the intensity of the sound. 


1. Sound originates at a vibrating body. 

2. Sound gives us information concerning: 1, the direction 
of the sounding body; 2, the number of vibrations per second; 
3, the intensity; and 4, the nature of the vibrations. 

3. Sound is a wave motion of the air. 

4. Sound waves are longitudinal. 

6. The velocity of sound in air is 331.7 -^- at 0° C. 


6. The velocity of sound in air increases 60 g^- for a rise in 
temperature of 1° C. 

7. Every elastic body has a natural period of vibrjttion. 

8. A body may be set into vibration by resonance when the 
impressed period is equal to the natural period of the body. 

9. The pitch of a note depends upon its vibration number. 

10. The vibration numbers of the notes of the piano are 
related to one another by the simple ratios 1: 2: 3, etc. 

11. The length of a sound wave in air depends on the vibra- 
tion number of the source. 

12. The vibration number of a stretched string is inversely 
proportional to the length and the diameter of the string; and directly 
proportional to the square root of its tension and its density. 

13. The number of vibrations of an air column is inversely 
proportional to the length of the column. 

14. The vibrations of a tuning fork are of the simple harmonic 

15. The intensity of a sound wave is proportional to the square 
of its amplitude. 


1. What is the origin of sound? How is this known? 

2. Why are we led to suppose that sound is a wave motion? 

3. Why do we believe that sound consists of waves in air? 

4. How do we prove that this supposition is correct? 

5. How do we define the elasticity of air? 

6. What leads us to conclude that sound waves are longitudinal? 

7. How fast does sound travel in air at 0° C? Does its velocity 
depend on the temperature? 

8. Does the velocity of sound in air depend on the pressure of the 

9. Give some familiar examples of resonance, and show how the 
phenomenon helps us to prove that sound is a wave motion of the air. 

10. On what does the pitch of the note depend? How is this 
proved? What can you say of the lengths of the waves that start 
from two vibrating bodies of different pitch? 

11. On what four characteristics of a string does its number of 
vibrations depend? In what way does it depend on each? How are 
these relations determined? 

12. Are the transverse vibrations of a rod faster than the longitu- 
dinal? How do you know? 

SOUND 389 

13. How is an organ pipe set into vibration? What is the action 
of the air in it? ' 

14. How may we increase the intensity of the sound emitted by 
a vibrating string? How is this done in the piano, the violin, the 
guitar, the cornet, and the trombone? 


1. Consider a stretched violin or piano string. The string rests 
on supports near its ends. At these points the string is not free to 
move. Are they, then, similar to that end of the jumping rope (Art. 
301) which was tied to the tree? Will these points be nodes when the 
string vibrates? Will it vibrate in stationary waves? If the string 
vibrates as a single loop, like a jumping rope when being used to jump, 
what is the relation between the length of the string and the distance 
between the nodes of the stationary wave along it? How does the 
distance between the nodes in a stationary wave compare with the 
wave length of the wave? If L represents the length of the string, 
and Zi the wave length of the wave formed when the string vibrates 
in one loop, show that L = J Zj. Similarly, if I2 represents the wave 
length when the string vibrates in two loops; Zg, the wave length cor- 
responding to 3 loops; Z4, that corresponding to four loops; show that 
L = i Zi = IZ2 = i ^3 = t ^4, etc. 

2. Draw curves similar to CNR, Fig. 183, showing the appear- 
ance of a string of length L vibrating in 1 loop; in 2 loops; in 3 loops; 
in 4 loops. 

3. Consider a solid rod 1 m long, 5 cm in diameter, and clamped 
in the middle. Since the middle point is clamped, there must be a 
node there when the rod is set vibrating in stationary waves. Strike 
one end of the rod so as to set it vibrating transversely and watch its 
motion. Do the points at the ends vibrate with greater amplitude 
than those near the middle? If so, and the rod is vibrating in sta- 
tionary, waves with a node in the middle, where do the centers of the 
loops lie? In a stationary wave, is the distance between the centers 
of the loops equal to that between the nodes? What is, then, the 
relation between the length of the rod and the length of the stationary 
wave on it? 

4. In the case of the rod of problem 3, the distance from the end 
of the rod to the node was J of the wave length of the stationary wave. 
Where must the nodes lie if the ends of the rod are free and it vibrates 
so that the length of the rod is 'a whole wave length? In this case, 
must the middle of the rod be left free to vibrate as a loop? Draw 
diagrams showing the nodes on the rod when vibrating so that its length 
is i wave; f waves; f waves. If the rod is always left free at ita ends, 
can the ends ever be nodal points? 


5. In the open organ pipe, Fig. 189, the air is free to move at both 
ends of the pipe. When the air column vibrates in this way, where 
will the node lie? What will be the longest stationary wave which the 
column of air in the pipe can form? If you draw diagrams showing 
the positions of the nodes when the air column is vibrating so that its 
length = J wave length, j wave lengths, etc., will these diagrams 
differ froni those of problem 4 in any way except that the vibrations 
of the particles of air are along the pipe instead of transverse, as the 
vibrations of the rod's particles are? 

6. The longest wave of an open organ pipe is twice the length of 
the pipe.^ Sound travels with a velocity of 1120 ^- at 15° C. What 
is the length of an organ pipe that gives the tone middle c {ut^ of 256 
vibrations per second? 

7. The longest open pipes in church organs are 32 ft. long. What 
is the pitch of the tone given by one of them?- 

8. We have seen that in an open pipe the free ends are always the 
places where the air particles vibrate with greatest amplitude, i.e., the 
open end always corresponds to the middle of a loop. If a pipe is closed 
at one end and open at the other (a stopped pipe), will the closed end 
be a node? Then how does the length L of the stopped pipe compare 
with the length l^ of the stationary wave? If you blow very gently 
across the mouth of such a closed tube, it sounds its fundamental 
or lowest tone; but if you blow harder it gives a higher tone. Does 
this indicate that another node has been formed, so that the air col- 
umn is vibrating in shorter stationary waves? Diagram the condi- 
tion of the air when there are two nodes, one, of course, at the closed 
end and the other between the two ends. If I2 now represents the 
length of the stationary wave, what fraction of Z2 is L? Blowing still 
harder across the open end, you may get a still higher note, which 
corresponds to stationary waves when there are three nodes, including 
the one at the closed ends. Diagram this condition of the air column, 
and state the relation of L to Z3. 

9. How long must an open pipe be in order that the longest sta- 
tionary wave in it shall be equal to that in a closed pipe 20 cm long? 

10. Can you compare the stationary waves on a rod, clamped at 
one end and free at the other, with those of a closed organ pipe, and 
show that L = \li = m = \lzy etc.? Show, by diagrams, where the 
nodes ought to be. Clamp a long, flexible, and elastic rod in a vise 
and see if you can make it vibrate transversely in 1, 3, and 5 half -loops. 

11. The numerical value of the elasticity of water is found to be 
205 X 107, its density 1 gm per cm^; what is the velocity of sound in it? 

12. When a tuning fork vibrates, its center of gravity remains 
at rest. How must the prongs move with reference to each other in 
order that this may be true? 

SOUND 341 

13. When the two prongs of a tuning fork are approaching each 
other while vibrating, they compress the air between them, thus start- 
ing a condensation in the wave. At the same instant is the air on the 
outer sides of the prongs compressed or rarified? If the fork thus 
starts a condensation and a rarefaction at the same time, why do not 
the two destroy each other^s effects so that we hear no sound? Hold a 
vibrating tuning fork near your ear, turn it about its long axis, and 
see if you can find any positions in which no sound is heard. If you 
find them, explain their presence. 

14. What is an echo? Suggest a way of determining approxi- 
mately, with the aid of a watch, the distance of a hill which gives an echo. 

15. Caii you prove by geometry that when sound spreads out 
from a small source, the intensity of the energy received on one cm^ of 
surface is inversely as the square of the distance? If you can do this, 
explain the use of speaking tubes and megaphones. 


1. Have you ever noticed the tones given by telegraph wires when 
the wind is blowing across them? How do these tones arise? Make 
an iEolian harp and put it in your window. 

2. If you have a flute, measure the distance from the mouthpiece 
to the hole that gives a certain tone and calculate the vibration num- 
ber of the tone. 

3. Perhaps you have noticed that when a rapidly moving loco- 
motive is whistling as it passes you, the pitch of the whistle changes 
at the instant when it reaches you. Does the pitch rise or fall while 
the train approaches? While it recedes? Can you apply your knowl- 
edge of the composition of motions to explain why this is so? Suspend 
an electric bell by wires from 10 to 30 ft. long, connected with a battery 
and push button. Swing the bell through a long arc and keep it ring- 
ing. What changes occur in the pitch? Why? 

4. The vibrations of organ pipes are well presented in Sedley Tay- 
lor, Sound and Music (Macmillan, New York). You will also find a 
great deal of interesting information about sound and music, and about 
fog signals, in Tyndall On Sound (Appleton, New York). 

5. For much information in very concise form, see Jones's Heatj 
Lightt and Sound (Macmillan, New York). Blaserna's Sound and Music 
is also good (Appleton, New York). For home experiments, see 
Mayer's Sound, and Hopkins's Experimental Science. 


326. Development of the Musical Scale. The first impor- 
tant problem concerning the musical scale is that of finding why 
we have selected certain particular pitches and put them together 
in a certain way to form the gamut of the piano. From the dis- 
cussion in the last chapter, it appears, that within certain wide 
limits, strings may be made to execute any number of vibrations ; 
and, therefore, with a large number of strings differing from one 
another in diameter, length, and tension, such as we have in the 
piano, we are able to produce a series of tones whose vibration 
numbers shall be related to one another in almost any way that 
we may choose. 

In the preceding chapter we proved, with the help of the 
punched disc, or syren (Art. 318), that the vibration numbers of 
the familiar notes, dq, fa^ sol, do, were related by the simple 
ratios f , f , f . It therefore becomes of interest to try to find 
out why we pick out a certain particular set of notes whose vi- 
brations are related to one another in such a simple and definite 
way. The answer to this question is in one way very simple, 
and in another very complex; but before we can answer it, we 
must find out what the relations between the numbers of vibra- 
tions of the different notes of the piano scale are, i.e., we must 
discover the manner in which that scale is constructed. 

The history of music helps us here; for from it we learn that 
mankind has not always had a musical scale, and that different 
peoples select different scales. We, for example, would find it 
difficult to recognize the productions of a Chinese orchestra as 
music. But even nations of our own type of civilization have not 
always had harmony as we now know it. The music of the early 
centuries of our era sounds harsh and ofttimes discordant when 
compared with modern compositions. Thus we learn that the 



present musical scale was not used in early times, and ihajt it has 
gradually developed into its present form, this form having been 
reached during the 16th century. Since Johann Sebastian Bach 
was the first who composed masterpieces in the modem scale, 
he is often called the '* father of modern music." 

327. The Eelated Triads. Go to the piano and play the 
two notes, middle c and g. Together they form a compound 
tone that pleases us, so we call it harmony. But this combination 
of c and g does not sound rich and full. We like the eflPect better 
when we add the note e and play together the three notes c-e-g. 
This combination of three notes satisfies us somehow; and if we 
strengthen the efiFect by playing also the octave of some or all 
of the three notes, we are still better pleased. Since the com- 
bination of these three notes produces such an efiFect on us, we 
make great use of it in musical compositions. We call this com- 
bination, i.e., the combination do-^mi-sol, a major triad. 

Now, although the major triad is a pleasing combination, it 
becomes monotonous when played continuously. Hence," we 
must seek for other triads for variety. When we try various 
other triads on the piano, we find that there are two others that 
seem to harmonize with the first. Thus, if we play c-e-g, g^b-d, 
c-e-g, we recognize that the two triads are in some way related; 
similarly, if we play c-e-g, f-a-c, c-e-g, which are recognized as 
the familiar amen at the end of hjonns. If now we play the 
three triads thus found in succession, viz., c-e-g, f-a-c, g-b-d, 
c-e-g, we perceive not only that we have played a pleasing suc- 
cession of chords, but also that we have been left with a sense of 
repose. We know that the piece has ended and we are satisfied. 
Therefore we conclude that in some way these three triads define 
a scale or key. These three triads, which together define a scale, 
are called the tonic {do-mi-soJ), the dominant {soUsi-re) and 
the SUBDOMINANT (Ja-la-do) triads. 

328. The Vibration Nnmbcrg. Having thus discovered that 
these three triads define a scale and that we select them solely be- 
cause they please us and give us a sense of harmony and repose, let 


us next find out if there are any numerical relations between the 
vibration numbers of the notes that compose them. This may 
be done in a number of ways, but is accomplished most easily 
by taking a string of given substance and tension and finding how 
its length must be changed in order to produce the tones of the 
triad (c/. Art. 320). The experiment is easily performed with 
a guitar, banjo, or mandolin string; for we have but to measure 
the length of the string from the nut to the bridge and then meas- 
ure the distance from the bridge to the frets that give the tones 
mi and soL When we do this, we find that these lengths are 
related to the lengths of the string by the ratios \, f, |, i.e., if 
the whole string gives the note do, the note mi is given by | of 
the string, and the note sol by § of that length. But since the 
vibration numbers of strings are inversely proportional to the 
lengths of the strings, we see that the vibration numbers of the 
notes of the triad are related to one another as are the ratios {, f , 
f ; or, what amounts to the same thing, by the ratios, 1, f , }. Thus 
we prove that the vibration numbers of the notes in a triad are 
related to one another as are the simple ratios 1, f , J. 

). The Major Scale. If we assunie that the note c exe- 
cutes 24 vibrations each second, we see that the numbers of vi- 
brations of the three notes of the triad c-e-g are 24 X 1 = 24, 
24 X f = 30, 24 X i = 36. What will then be the vibration 
numbers of the notes of the second triad g-b-dt Since the lowest 
note of this triad is the same as the upper note of the other, the 
vibration numbers of its notes will- clearly be 36 X 1 = 36, 36 X | 
= 45, 36 X f = 54. To get the corresponding vibration lum- 
bers for the triad /-a-c, we note that it contains a note c which is 
an octave above the c in the triad c-e-g. But we learned in the 
last chapter that the octave executes twice as many vibrations a 
second as the lower note. Therefore the number of vibrations 
of the upper c is 48. Since this note is the third note in this triad, 
this number corresponds to f . Hence, the note / in this triad will 
be 48 X I = 32, and the note a, 32 X f = 40, or 48xf =40. 

We thus find the following vibration numbers for the notes 
in these triads:' c = 24, e = 30, gr = 36, 6 = 45, d = 54, c = 48, 


a = 40, / = 32. We note that all the numbers lie between 24 
and 48, excepting d, which corresponds to 54. In order to bring 
this number within the desired octave, we transpose this note 
down one octave and thus find the vibration number of the lower 
d to be V" = 27. We now arrange these notes in the order of 
their vibration numbers and get the series: 

c d e f g a b c 

24 27 30 32 36 40 45 48 

On inspecting this series we find that it contains all the notes 
of the musical scale which correspond to the white keys on the 
piano, i.e., the notes do, re, mi, fa, sol, la, si, do. But this series 
of notes is composed only of those notes which appear in the 
three triads which we have found necessary to define a scale 
or key. Hence we see that the musical scale is selected as it is, 
in order that it may contain all the notes necessary for the pro- 
duction of the three major triads which we have selected for the 
reason that they produce, when played together, a feeling of 
satisfaction and repose. It appears, then, that these particular 
notes have been adopted for a musical scale because something 
connected with our perception of sound leads us to pronounce 
certain combinations of tones harmonious or pleasing, and others 
discordant or disagreeable. 

It is interesting to observe that the ratios of these numbers 
can always be expressed as the ratio of small whole numbers. 
This fact was clearly perceived as long ago as B.C. 525 by Pythag- 
oras, who propounded the problem in the question. Why do 
we call a combination of tones harmonious when the vibration 
numbers of the component tones are related to one another by 
the ratios of simple whole numbers? This problem of Pythag- 
oras remained without answer for over two thousand years. 
Helmholtz, in 1871, finally solved it. But before passing to his 
solution of it, we must complete the definition of the musical 
scale, for we have only found the ratios that exist among the 
vibration numbers that correspond to the notes of the white keys 
of the piano. We have yet to find out why there are black keys 
also. Further, we must determine the actual numbers of vibra- 


tions of the diflPerent notes, for the numbers that have just 
been given express merely their ratios. 

330. The Complete Scale. The necessity for the black keys 
becomes apparent when we wish to play a set of triads beginning 
with e instead of with c. Since the vibration number of the note e 
is represented by 30 in the table just given, we see, by applying 
our ratios 1, f, J, that the relative vibration numbers of the 
notes in the triad beginning with e would be 30 X 1 = 30, 30 X 
I = 37J, 30 X f = 45. The note represented by 45 already 
exists in the scale at 6, but we have no note corresponding to 
37J. Since this number falls nearly half-way between 36 and 40, 
it has been found necessary to add another note to our scale about 
half-way between g and a. This note is called g sharp, and it 
is clearly added to enable us to play scales that begin on e instead 
of on c, thus increasing the number of scales that can be played 
on the instrument. Similarly, if we wish to begin a scale on a, 
which is represented by 40, the second note of the triad would be 
represented by 40 X f == 50, or by V" = 25; and the third by 
30 X f = 60, or by Y = 30. Now, the note 30 already exists 
in our series, but an extra note corresponding to 25 has to be added 
between 24 and 27. This note is called c sharp. Similarly, by 
figuring the numbers of vibration of the triads that begin on the 
notes d and 6, we find it necessary to add other notes between / 
and g and between d and e. The reason for adding the black 
keys is therefore apparent. They enable us to play scales that 
begin on notes other than c. 

But as we proceed with this addition of notes our series soon 
becomes very complex. For when we construct the triad that 
begins on d, we find the corresponding numbers to be 27-33i- 
40^. We can supply the note represented by 33 J by adding 
one between / = 32 and g =36. But the number 40^ does not 
agree with a = 40, though it comes pretty near it. Similarly, 
when we come to supply the triad which shall have c = 48 for 
its middle note, we find the numbers 38f-48-57|, or reducing 
the latter one octave 28f-38|-48. Now, we have already 
added one note between 36 and 40, viz., g sharp = 37^, and 


this differs slightly from the one that now appears to be neces- 
sary, viz., 38|. If we carry this process of working out triads 
further, we find that very many notes would have to be added 
in order to make it possible to play scales which begin on all the 
notes of the scale of c. A keyed instrument of the piano type 
would require about 70 keys to the octave and would soon become 
too complicated to manage. Yet every one knows that we can 
play all the different scales on the piano. How, then, is the diffi- 
culty avoided? 

331. Tempered Scale. The answer is simple. We insert 
extra notes which do not exactly satisfy either of the required 
conditions, i.e., when two notes have vibration numbers that are 
very nearly equal, we take an average note and let it do for both. 
Thus, instead of having a note 40 and another 40^, we make one 
note do for both by tuning the strings so that the number of vi- 
brations shall correspond to about 40^; similarly with the other cases. 
We do not have on the piano a note 37^ and another 38|, but one 
note corresponding to about 38, etc. By doing this .we do not 
produce the triads in perfect tune, but we approach nearly enough 
to perfect tune for all practical purposes. The scale in which 
these adjustments have been made is called a tempered scale, 
to distinguish it from a scale in which the notes are related by the 
correct ratios. 

All keyed instruments, like the piano, the organ, the clarionet, 
in which each key corresponds to a note of definite pitch, must be 
tuned to the tempered scale. On the other hand, stringed in- 
struments, like the violoncello and the violin, may be played in 
the pure scale. It is for this reason that many musicians find the 
piano music disagreeable. It is related of Handel that he could 
not bear to hear music played in the tempered scale, so that he 
had constructed for himself an organ which had keys for every 
one of the notes demanded by the theory. A musician like Handel 
might be able to play upon a keyboard as complicated as this, 
but less gifted individuals would evidently be able to do nothing 
with it. 

In tempering the scale, what method is employed? Do we 


simply guess at the probable location of the notes desired, or do 
we adopt a fixed principle which shall render the departures 
from accurate tuning as small and as evenly distributed as pos- 
sible? Evidently the latter procedure is the only strictly scien- 
tific one. The principle which is adopted is that of dividing 
the interval of the octave into twelve equal parts. Since the 
numbers that represent the series of notes express ratios merely, 
this division into twelve equal pa,rts must be done by finding a 
number such that, if we multiply 24 by it twelve times in suc- 
cession, the result will be 48. This number has been found to be 
1.059, and if we multiply 24 by it twelve times we get for the 
numbers that correspond to the notes on the piano scale those 
indicated in the following table. The numbers that indicate the 
true intonation are added in order to make clear just how great the 
departures of the tempered scale from the theoretically correct 
one are: 






c» d\> 






d# eb 


e , 











gf# Ob 





a# 61- 








If we examine these numbers we see that the notes d and g 
are but slightly out of tune; while some of the others, like a, are 
badly so. However, for an instrument like the piano and the 
organ, in which the notes are fixed, this distribution of error 
seems to be the best that can be made without unduly increas- 
ing the number of keys. 


332. Standard Fitch. One more factor remains to be deter- 
mined before the scale is completely defined. The numbers that 
have been given express merely the ratios between the vibration 
numbers of the different notes. In order to fix the scale com- 
pletely, therefore, we must state how many vibrations some par- 
ticular note gives. Two definitions of this sort are in common 
use. The physicist says, I will define the note middle c to be 
that note which executes 256 vibrations per second. The num- 
bers of vibrations of the other notes of the scale may then 
be found by multiplying 256 by the ratios given in Art. 329. 
Thus, the vibration number of middle g is 256 X f J = 384, 
etc. The musician, however, uses a different absolute pitch, 
for he defines the note a, which is the a of the violin, to be 
that note whose number of vibrations is 435 per second. This 
definition, viz., a =435, is called the international standard 


333. Forced Vibrations. Having thus found what the numer- 
ical relations are between the notes of the musical scale and learned 
that they have been so chosen because of something connected 
with our perception of sound, we will now proceed to see if we 
can find out what that something is. Although our ears are 
complicated structures, yet the physical principle that finds ap- 
plication in their operation is rather simple. It is none other 
than that of resonance, with which we have become familiar in 
Art. 314. 

We there learned that a strict agreement between the natural 
period of the body and the impressed period of the wave is neces- 
sary for the production of resonance. While this is true in many 
cases, it is not always true, for light flexible objects will often 
vibrate by resonance when the impressed period coincides only 
approximately with the natural period. In such cases the vi- 
brations will be most violent when the agreement between the 
periods is exact, and Will diminish in intensity as the difference 
between those periods increases. The vibrations produced by 
resonance when the natural and the impressed periods are not 
the same, are called forced vibrations. 


334. The Ear. The next step in solving the riddle of Pythag- 
oras is indicated by the question, does the ear perceive sound 
by resonance? Are there in the ear a series of bodies whose 
natural periods of vibration are different, so that they would 
be set vibrating by different impressed periods? The answer 
to this question can, of course, be found only by an anatomical 
investigation of the construction of the ear. This has been done, 
and it is found that there are in the ear a large number of jfine 
fibers of different lengths. These fibers are fastened at one end 
to a membrane which is inside a tiny cell that looks like a small 
snail shell, and is therefore called the cochlea. The cochlea is 
full of liquid, so that the fine fibers — called fibers of corti, 
after their discoverer — are surrounded by the liquid. The mem- 
brane in wliich these fibres end is connected to the auditory 
nerve which carries the sensation to the brain. The arrangement 
of the cochlea and the other parts of the ear is shown in Fig. 

The main thing that interests the physicist in the construction 
of the ear is the presence of this series of fibers of Corti; for these 
may be a series of bodies which have 
natural periods of vibration, and which 
may, therefore, be set into vibration 
by resonance by notes of different pitch. 
And this is what we believe them really 
to be — a veritable set of resonators, 
each tuned to one of the notes which 
we are able to distinguish within the 
range of the musical scale. But how 
many would that be? Experiment tells 
us that we can not hear sounds whose numbers of vibrations are less 
than about 30 per second or more than about 30,000. So these 
little resonators in our ears must be tuned to notes that fall within 
this range. Attempts have been made to count them, and there 
are found to be about 3,000 altogether. Therefore there must 
be one for each difference of about 10 vibrations. It is probable 
that there are more than this in the middle of the aiusical scale 
and fewer at the outside limits; but however this may be, it is 


certain that there is not one for each diflPerence of one vibration. 
And yet we can detect difiFerences of less than this amount in 
pitches and can hear tones with all conceivable numbers of vi- 
brations within the range just mentioned. How is this possible 
if there is only one resonator in the ear for each difference of 10 

To account for this, Helmholtz holds that because these Corti 
fibers are flexible and light, they vibrate by resonance in response 
to notes whose periods are nearly the same as their own natural 
periods. If this is so, then a fiber whose natural period is y^^ 
sec will respond to impressed periods that lie, say, between jj^ 
and Yki^ sec. Hence we see that any particular note must affect 
several of the little resonators in the ear, acting most strongly on 
that one whose natural period is nearest to the impressed period. 

To sum up what we have thus far learned, we see that the ear 
contains a large number of tiny resonators (fibers of Corti) which 
are tuned to different notes throughout the range of audible tones; 
when a sound wave of definite period falls on these resonators several 
adjacent ones are set into vibration. This knowledge of the con- 
struction of the eAr is essential if we are going to understand at 
all the reasons for harmony and discord. 

335. Beats. We may now ask what sort of excitement 
of these fibers of Corti would be disagreeable. We can imagine 
a sort that would probably prove disagreeable by considering the 
similar case of light; for we all know well that a steady light is 
necessary for any comfort in seeing, while a flickering light, pro- 
vided the number of flickers is neither very great nor very small, 
is intolerable. May it not be that a flickering sound would be as 
intolerable as a flickering light? But what is flickering sound? 
Clearly one in which periods of sound and silence follow one 
another closely, just as the flickering light is one in which periods 
of light and darkness follow one another closely. Do we ever 
have flickering sounds? Let us see. 

Take two tuning-forks, or organ pipes, or other sources of 
sound whose vibration numbers differ slightly, one being, say 
greater than the other by one. Conceive them to be started at 


once in opposite phases. Then the two waves (Fig. 191) which they 
send out will start in opposite phases, and an observer will hear no 
sound. But since one of the waves is shorter than the other, 
and since they both travel with the same velocity, the phase of 
the shorter wave will gradually gain on that of the longer wave 
until' the phases coincide( a, Fig. 191). When this condition has 


Fig. 191. Beats 

been reached, the two waves add together their effects, and a 
period of loud sound results. If there is in each second one more of 
the shorter waves than of the longer, the loudest sound will occur at 
a, in the middle of the second. As the waves proceed further the 
shorter again outstrips the other in phase, until at the end of the 
second b, they are again opposite in phase and we hear no sound. 
If the difference in the vibration numbers of the two notes is 
2, then there will be two periods of silence in each second, and so on. 
If iVj and iVj represent the numbers of vibrations per 
second of the two notes, then N^ — N2 =" n will be the number of 
periods of silence in a second. Two tones that produce flickering 
sound in this way are $aid to give beats, and the number of 
beats per second is equal to the difference in the numbers of vi- 
brations of the two. 

336. Discord Due to Beats. Having thus found out how a 
flickering sound may be produced, let us see if the result is disa- 
greeable. The experiment may be tried in the laboratory by 
sounding together two organ pipes, or two tuning forks of slightly 
different pitch. They Jire also readily audible when two adjacent 
lower notes on the piano are sounded together. When they are 
slow they can be counted. When they get faster they become 
disagreeable, and when they become very rapid — more than 


about 30 beats per second —they fail to be distinguishable and the 
disagreeable sensation ceases. Therefore Helmholtz concludes 
that discord is due to beats, and that we call two tones discordant 
when their combination produces between four and thirty beats 
per second. 

But even with this explanation of discord we are still far from 
our goal. For if notes that give not over 30 beats per second are 
discordant, why do we object to the combination c-/# and prefer 
the harmony, c-^? Since the numbers of vibrations of c, /# and 
g a*re 256, 376, and 384, the numbers of beats in these .two cases 
are 120 and 128. Since these numbers of beats both fall outside 
of the disagreeable limit of 30, why should we judge one of the 
combinations of tones harmonious and reject the other as dis- 
cordant? Before we can answer this question we shall have to 
discover the reasons for differences in quality between the tones 
of different musical instruments. As this inquiry is somewhat 
long, we shall devote the next chapter to its study. 


1. The musical scale has not always existed in its present 

2. The notes whose vibration numbers are related by the ratios 
I, f , J are called a major triad. 

3. We choose the ratios f, |, J for the triad because we find 
by experiment that the combination of the corresponding notes 
is pleasing. 

4. There are three triads whose relationship is very close, viz., 
the tonic, the dominant, and the subdominant. 

5. These triads together contain all the notes of the musical 
scale, and therefore define the scale. 

6. The ratios of the vibration numbers for the notes of the 
major scale are expressed by the following numbers: 

c d e f g a b c 

24 27 30 32 36 40 45 48 

7. Intermediate notes have to be added to this scale if we wish 
to be able to play scales beginning on notes other than c. 


8. This addition of intermediate notes makes tempering 

9. Pianos and organs are tuned to the tempered scale. 

10. The physicists' standard of pitch is c = 256, while the 
musicians' standard is a = 435. 

11. The ear contains a series of resonators called fibers of 
Corti, whose natural periods lie within the limits ^and jt^dd sec. 

12. A Corti fiber is affected by a vibration even though the 
agreement between its natural period and the impressed period 
is not exact, so that one impressed period produces vibrations 
in more than one fiber. 

13. Two notes of different numbers of vibrations produce 
beats. The number of beats per second Is equal to the difference 
between the numbers of vibrations of the two notes. 

14. When two notes produce from 4 to 30 beats per second, 
the sound flickers and we call it discord. 


I. How can we find the ratios of the numbers of vibrations of 
the three notes in a triad? What are those ratios? 

. 2. Why do we say that the tonic, dominant, and subdominant 
triads are related? Why do they define a key? 

3. How do we derive the relative numbers for the second and third 
triads from the first? What are these numbers? 

4. What is needed for defining a scale in addition to these numbers? 

5. Why is it necessary to add the black keys to the piano key- 

6. Why do we temper the piano notes? Upon what principle is it 

7. Will a continuous force set a body vibrating? What sort of 
force will? 

8. What relation must exist between the period of the impressed 
force and the natural period of a body in order to produce sustained 
vibration? When this relation is not exact, can resonance occur? 

9. Can a stretched string detect- a sound? If so, when? 

10. What do we suppose to be the action of the fibers of Corti in 
the ear when a sound is heard? 

II. Do we believe that one or that more than one fiber of Corti 
vibrates by resonance when a note of definite period is impressed on 
the ear? State the reasons for your answer. 


•12. What sort of sound is disagreeable to the ear? How is such a 
sound produced? 

13. If two notes have relatively Ni and ^^2 vibrations per second, 
how many beats will they produce when sounded together? 

14. Why are beats disagreeable only when we have more than 4 or 
less than 30 per sec? 


1. Suppose a banjo string to be 90 cm long from the bridge to the 
nut. Calculate the distances from the bridge to the frets that give 
the various tones of one octave of a major scale, using the ratios of 
the vibration numbers as given in Art. 329. 

2. Harmonics are produced on a violin string by lightly touching 
the string at points i, J, J, i, etc., of the length of the string. This 
forces a node on the strirtg at the point touched, and causes the string 
to vibrate in 2 loops, 3 loops, 4 loops, etc. What are the tones that 
may be obtained from the string in this way? 

3. Pythagoras proposed to determine the tones of a musical scale 
by taking only intervals of a fifth (do-sol), starting from a given note. 
Beginning with c = 24 vibrations, find the number corresponding to 
its fifth, g. Then find the fifth of gr, by multiplying by the ratio |. 
Continue the process, reducing eaqh number that falls outside the 
limits 24-48, by i, and see if you can find out why such a scale is 
impracticable. This process amounts to raising } to the power n. 
Is I commensurable? Will (I)** be commensurable? Will you ever 
by this process reach a note that is an octave of the one from which 
yoii started? 

4. Calculate the vibration numbers of the triad beginning on a = 40 
and those of the one whose middle note is / = 32. What numbers 
must be added to those in the scale in Art. 329 to enable you to play 
these triads? How nearly can you produce these triads with the 
tones of the tempered scale. Art. 331? Repeat the calculation for 
the triad beginning on 6 = 45 and the one whose middle note is g =• 36. 
How do these triads fit the tempered scale? 

5. A string under a tension 0+-600 gms force, gives middle C (Ci). 
Under what tension will it give J^i? Gi? C2? 

6. A string 60 cm long and 0.5 mm in diameter gives Fj, what 
must be the diameter of a string of the same length and under the same 
tension in order that it may give A^? C2? A2? 

7. A violin bow is drawn across the top of a narrow strip of spring 
brass 10 cm long, and it gives a certain tone, say C2. Consider it as a 
rod: to what length must it be reduced in order to give the octave C3? 

8. A steel rod 80 cm long, clamped at the middle and rubbed with 
a rosin cloth gives the tone A3. What changes in its length will cause 


it to give successively the other seven notes of the scale beginning on 
this note? 

9. An organ pipe is 8 feet long. What must be the length* of a 
pipe, all other things being equal, that will give the fifth below the note 
given by the first? The octave above? 


1. If you have a banjo, a guitar, or a mandolin, measure the dis- 
tances from the frets to the bridge and see if they satisfy the laws of 
vibrating strings and the vibration numbers of the major scale. Can 
you find out whether the frets are tuned to the tempered or the pure 

2. If you play a violin, or have a friend who does so, play the har- 
monics and get your friend to measure the distance of your finger 
from the nut or bridge. Is this distance always an aliquot part of the 
length of the string? Can you recognize the pitches of the har- 
monics produced, using, if necessary, a piano to assist you? Do the 
vibration numbers of these notes "check up" with the lengths of the 

3. Can you find out, with the help of a standard tuning fork, whether 
your piano is up to concert pitclj? Are all pianos and organs really 
tuned to the same pitch? When your piano is being tuned, consult 
the timer and find out if he uses beats to determine where the two 
strings are in tune. Can you find out how organs are tuned? 

4. Can you make a musical instrument that will play the scale, 
by driving pieces of knitting needles into a board? If you succeed, 
measure the lengths of these rods, and see if they follow the law for 
transverse vibrations as stated in Art. 321. Examine a musical box 
and see if it is made on this principle. 

5. See if you can cut a long rod of dry, elastic wood into pieces of 
such lengths that they will play the scale when you lay them across 
two wedge-shaped sticks and strike them with a light hammer. Look at 
a xylophone in a music store and see if this principle applies to it. Does 
the tone depend on where the supports are placed ? 

6. How are the vibrations of a violin string communicated to the 
body? What part has the air in the body in producing and sustaining 
the tone? 

7. Try to get a loud sound from a wire stretched between two iron 
gate posts. Does the result indicate that the air is set into vibration 
by the string of a violin, or is it the body that does this work? 

8. What can you find, by examining the sound-board of a piano, as 
to the way in which it is adapted in shape and construction so as to 
give resonance to notes of various pitches? 


337. Wave Shape and Tone duality. We shall devote this 
chapter to the discussion of the last point in our investigation into 
the physical basis of harmony and discord. This point is in- 
volved in the question, why do we call notes like c and /# discord- 
ant when they produce, when sounded together, as many as 120 
beats per second, while no discord results when the beats are less 
than 4 or more than 30? In order to answer this question, we 
must recall some of the facts presented in the preceding chapters. 

First, waves bring us information as to: 1, the direction of the 
source; 2, the number of vibrations of the source; 3, the intensity 
of vibration of the source; and 4, the nature of the vibration of 
the source. We have further identified these four kinds of in- 
formation with the characteristics of waves, as follows: 1, the 
direction of propagation depends on the direction of the source; 
2, the length of the wave is connected with the number of vibra- 
tions or pitch of the source; 3, the amplitude of the wave is derived 
from the amplitude of the vibration of the source; and 4, the 
shape of the wave varies with changes in the nature of the vibra- 
tion of the source. 

When we apply these facts to sound, we see that the direction 
of propagation tells us of the direction of the source of sound, 
that the wave length tells us of its pitch, and the amplitude tells 
us of its intensity. But what information do we derive from 
differences in the shape of the sound waves? How do we detect 
these differences in shape? Since the only characteristic of 
sound remaining for determination is its quality, and the only 
characteristic of waves remaining undetermined is their shape, 
we may conjecture that our perception of differences in qualities 
in sound is dependent on our perception of differences in the 
shape of the sound waves. 




Let us then adopt the hypothesis that tone quality is connected 
with wave shape, and see whether it will help us in getting the 
answer to the problem before us. The first step in the discussion 
of this question is that of determining how differences in the 
shapes of waves are produced. This we have already done, for 
we have learned that waves of complex shape are produced by 
adding together simple homogeneous waves of different lengths, 
amplitudes and phases (cf. Art. 300). Hence we can conceive 
that a sound wave of complex shape would result from the addi- 
tion of two or more simple sounds differing from one another in 
pitch and intensity. 

Fia. 192. The Vibratinq 


338. The Vibrating Flame. That this conception corresponds 
with the facts may easily be shown by experiment. We have but 
to devise a scheme for rendering the motion of the air particles 

visible, and then to bring several sources 
of simple homogeneous waves together, 
to see if the resultant motion of the air 
does not indicate that we have a com- 
plex wave. ^Probably the simplest 
method of doing this is th^ following: 
A thin rubber membrane AB (Fig. 192) 
is mounted between two rings of wood. 
A flexible tube C leads the sound waves 
up so that they can act on one side 
of this membrane. The membrane will then follow the vibra- 
tions of the air in the tube C. On the other side of the membrane 
is a small gas chamber D. Illuminating gas flows into this cham- 
ber at F and bums at the jet E. Whenever the membrane AB 
vibrates, the gas in D will vibrate also; and this will cause the 
flame to vibrate, the tip of the flame following roughly the vibra- 
tions of the membrane. Since the vibrations of sound are too . 
fast to be observed by the imaided eye, we have to observe 
the flame in a nrirror which is kept in rotation. The apparatus 
ready for use is shown in Fig. 193. When thus observed in the 
rotating mirror and no sound is acting, the image of the small 
fliame appears to be drawn out into a straight band of light; but 



when a train of sound waves is allowed to strike against the mem- 
brane, this band is no longer straight, but its upper edge assumes 
a wave-like form which must correspond closely in shape to that 
of the waves impressed on the membrane. 

Let us first send in sound waves from the tuning fork (Fig. 193), 
which, as we have learned, produces waves that are nearly homo- 
geneous; the appearance of the flame in the rotating mirror will 
be then shown in the top band in Fig. 194. Using a second tuning 

Fig. 193. Apparatus for Observino Vibrating Flames 

fork, an octave below the first, the appearance of the* flame will be 
that shown in the second band in the figure. If now we send in 
the waves from both these tuning forks at the same time, the 
appearance of the flame in the rotating mirror will not be the 
same as before, for we have now added together two waves of 
different periods, and therefore have a complex wave. The 
result is shown in the third band in the figure. Referring to curve 
R in Fig. 181, page 309, we see that the shape of the top of the 
band of light agrees roughly with the shape of the curve there 



obtained as the resultant of two waves, one of which had half the 
period of the other. 

We thus prove that two or more simple homogeneous sound 
waves add themselves together just as other waves do, and pro- 
duce resultant waves of complex form. 

339. Are Musical Tones Complex? The question then arises, 
are the waves sent out by piano strings, violin strings, or the human 
voice simple homogeneous waves, or do they have complex forms? 

If they are complex, do two 
complex waves of the same 
period, but corresponding to 
tones of different quality, 
have different shapes? The 
answer to this question is 
easily obtained from obser- 
vations with the little vibrat- 
ing flame. For if one of us 
sings into the flexible tube C 
at the same pitch the vow- 
els a, o, the appearance of 
the flame in the rotating mir- 
ror will be as shown in the 
fourth and fifth bands in 
Fig. 194. This result is a 
most striking confirmation of 
the hypothesis that tones of 
the same pitch but of differ- 
ent qualities have wave forms of different shape. 

The tones from strings, organ pipes, and other musical instru- 
ments, when analyzed in this way with the vibrating flame, show 
differences in wave form corresponding to their different qualities. 
But waves of different shape are produced by compounding simple 
waves in various ways. Therefore we see that a complex tone 
must be produced by the addition in various ways of simple tones, 
and that the quality of the complex tone depends on the way in 
which the various simple tones happen to be brought together. 





Fig. 194. Appearance of the Vibrat- 
ing Flame 


340. How Musical Tones axe Possible. It is easy to see how 
complex waves may be produced by the addition of two or more 
simple homogeneous waves of different lengths which originate 
from different sources, as from two or more tuning forks. But 
we have just learned that a single vibrating body, like the human 
vocal organ or a musical instrument, produces such complex 
waves. How can a single vibrating body produce several different 
vibrations at the same time? And if it does do so, are there any 
relations among the different vibrations which are thus produced 
at the same time? Recall the jumping rope (Art. 301). We 
learned that it may vibrate in one loop, in two loops, in three loops, 
depending on how rapidly it is turned. Similarly, a stretched 
string (Fig. 195) may vibrate in one loop, in two loops, in three 

Fig. 195. The String May Vibrate in Three Loops 

loops, etc. Since in this case the string is stretched with a con- 
stant force, and since the lengths of these loops are 1, i, J 
the length of the string, etc., the vibration numbers of the notes pro- 
duced are related by the simple ratios, 1: 2: 3, etc. Suppose 
it were to vibrate in several of these ways at once, what would 
be its shape? We can find out by adding together the com- 
ponent vibrations as in Art. 300. Thus, if we conceive the string 
to be vibrating in one and in two loops at the same time, and that 
the amplitude of the 2 loops is only half that of the 1, the result 
is shown at R in Fig. 181. If now, in addition, it is vibrating in 
three loops, and the amplitude of the 3 is but J that of the 1, we 
add the 3 loop wave to the resultant of the other two and obtain 
the curve shown at iJ' in Fig. 181. Similarly, by adding the 4 
loop vibration with J the amplitude of the 1, and the 5 loop vi- 
bration with ^ the amplitude of the 1, we get the resultant shown 
at P and Q, Fig. 196. If we continue this process of adding the 
curves that correspond to greater numbers of vibrations, each 



with a correspondingly smaller amplitude, the resultant becomes 
more and more like the curve at R in the figure. 

We thus see that a string may vibrate in all these ways at 
once if it can take the shape shown at R. But can strings take 
that shape? What would be the shape of a stretched string if a 

piano hammer had hit it 
at a point near the end? 
How does the violin bow 
act? Does it not pull the 
string into a shape similar 
to that shown in the figure 
until the tension of the 
string becomes great 
enough to overcome the 
friction of the bow? Then 
the string flies back. Or, if we merely pick the string with a sharp 
point, as in the case of the mandolin, we bring the string to the in- 
dicated shape and then let it go. So we see that a string may be 
made to take the indicated shape, and therefore we may infer that 
it can send out a compound wave similar to that composed of a 
number of vibrations whose periods are related to one another, 
as 1: 2: 3, etc., and whose amplitudes continually decrease. 

Fig. 196. Complex Waves op a String 

341. Fundamental and Overtones. We must now distin- 
guish between the tones thus produced. For this purpose we 
call the tone that corresponds to the vibration of the string in one 
loop the FUNDAMENTAL. It has the smallest number of vibrations 
and is most intense of them all. The other tones are called over- 
tones, or harmonics. Since the number of vibrations of the string 
in two loops is twice that of the string in one loop, the first over- 
tone will be the octave of the fundamental. Similarly, the second 
overtone is related to the first overtone by the ratio |, and will 
therefore be to the first as ^ to c in the musical scale. This 
interval is called a fifth. Similarly, the third overtone is related 
to the first overtone by the ratio |, and will therefore be an octave 
above the first, etc. 

There are several interesting things about these overtones. 


In the first place, the quality of a complex tone evidently depends 
on which overtones are present and how strong each is; for we 
have shown that notes of different quality produce complex waves 
of different shapes, and also that differences in shapes of waves 
are produced by differences in the number and strength of the 
simple waves of which they are composed. In the second place, 
we can show that a simple wave produces resonance in a body 
whose natural period agrees \iith that of the simple wave, not 
only when the simple wave exists alone, but also when it is a 
component of the complex one. 

342. Overtones of Piano Strings. The simplest way of showing 
this is the following: Press a key, say middle c of the piano, 
gently, so that the hammer does not strike the string, but so that 
the muffler is lifted. The string will then be free to vibrate by 
resonance. Then strike the key C, an octave below, and let it 
rise again so that the muffler stops the vibrations. The tone 
middle c will be heard gently humming in the piano. But we 
have just learned that the lower C contains the upper c as its 
first overtone; so we see that, since the overtone c exists as part 
of the compound tone C, the string whose natural period agrees 
with that of this overtone is set into vibration by resonance. Simi- 
larly, if we prcss the note g above middle c so as to lift its muffler, 
and then again strike the lower C, the tone g will be heard coming 
from the piano, showing that g exists as an overtone in the vibra- 
tions of C If, however, the experiment be tried with the note / 
above middle c, no tone will be heard from the piano after lower 
C has ceased to vibrate, because / is not an overtone of C, and 
therefore the vibrations corresponding to / do not exist in the 
complex tone emitted by C 

343. Helmholtz Eesonators. The experiment is even more 
striking if we use as resonators, not the strings of the piano, but 
a series of hollow brass spheres, whose volumes are such that the 
natural periods of the volumes of air in them coincide with the 
periods of different notes of the piano. Such a series of hollow 
spheres was used by Helmholtz in analyzing these complex tones. 


One of them is shown in Fig. 197. The little projection on one 
side is intended to be fitted into the ear, thus enabling it to detect 
very faint sounds whose periods agree with that of the resonator. 

344. How the Ear Perceives a Complex Tone. Let us now 

expand our conception of the resonance effect of a compound 

tone to include all the overtones at once. What effect will be 

produced on a series of strings, tuned to all the 

'^^ notes of the scale and free to vibrate, if we 

i. ^^k sound near them a compound note whose f un- 

m^* Jj^^^^ damental agrees with one of the notes of the 

^Bij^^^v scale? Evidently those strings that correspond 

^^^^^ to the overtones will be set into vibration by 

Fig. 197. Reso- resonance, while the others will remain at rest. 

NATOR 1 1 rt. 1. 

Thus we see that the etiect of a compound tone 
on such a series of strings, like those of a piano or harp, is similar 
to that produced by a skilled hand passing rapidly over the strings 
and touching gently those among them that correspond to the 
overtones and the fundamental note. 

We have learned that the ear contains such a series of strings 
or fibers, and so we may imagine that when a complex note falls 
on the ear, not all of these fibers are excited, but only those whose 
natural periods agree approximately with the periods of the fun- 
damental and the overtones of the note. 

We may now reach our final conclusion concerning discords; 
for since it appears that all musical notes are complex, and since 
5uch a note excites in the ear not only the fibers corresponding 
nearly to its fundamental, but also those corresponding to the 
overtones, it becomes clear that to obtain harmony between two 
notes we must avoid beats, not only between the fundamentals, 
but also between the overtones. Therefore the complete answer 
to our question as to the reasons for discord is, two tones are dis- 
cordant when either their fundamentals or any of their overtones 
produce beats which are more than 4 or less than 30 per second. 

It now remains for us to show that this principle will enable 
us to make clear why the interval c-g is more pleasing than c-/#. 
In order to do this we have merely to write out the numbers of 


Vibrations of the fundamentals and of the overtones and see 
whether we have such beats anywhere. These numbers are, for 
the three notes under consideration, 


















It thus appears that the discord between c and /# is due to the 
production of 16 beats by the second overtone of c and the first 

345. Belated Tones. This table brings to light another in- 
teresting fact concerning the notes c and ^r, viz., that some over- 
tones are common to both. We see that both have an overtone 
of 768 vibrations and another of 1536. Noting this fact, Helm- 
holtz calls such notes musical relations, i.e., he says that when two 
tones have two or more overtones in common, they are musically 
related. Such musical relationship must occur between notes 
whose fundamental vibration numbers are related by the simple 
ratios 1: 2: 3: 4: 5: 6, etc., because the vibration numbers of the 
overtones are related to those of the fundamentals by these same 

And so, at last, we reach the answer to Pythagoras's problem. 
It may be stated in many ways, but perhaps the simplest is the 
following: The numbers of vibrations of the overtones of strings 
and air columns are related to those of the fundamental by the sim- 
ple ratios 1: 2: 3, etc. Therefore the notes of the scale must be 
related by the same ratios in order to avoid disagreeable beats be' 
tween both fundamentals and overtones. 

346. Chimes. Do we ever use other sources of musical tone 
besides strings and air columns? We might answer, ''Yes," and 
cite, as an example, chimes of bells, which are justly reputed to 
produce a decidedly musical effect. But did you ever hear a chim^i 
of bells played in chords, i.e., more than one note at a time? Prob 
ably not, because the overtones of the bells are not related to th^ir 
fundamentals by the simple ratios 1: 2: 3, etc., and so when bells 


are played in chords, the effect is musically intolerable, because 
disagreeable beats occur between the overtones. 

Another interesting conclusion is, that fundamental tones 
which have no overtones would not be disagreeable, when the same 
fundamental tones with overtones would be so. This is easily 
shown to be true by sounding together two tuning forks with 
pitches c and /#, for instance, and comparing the effect with that 
produced by two organ pipes or strings of the same pitches. The 
combined effect of the forks is not at all disagreeable, while that 
of the pipes or strings is decidedly so. 

There are many other interesting and perplexing questions 
concerning tone quality and concerning harmony and discord. 
For example, how can we control the tone quality, as in the organ, 
where we make pipes whose tones resemble flutes, violins, horns, 
and even the human voice? How are the wonderfully different 
qualities of the human voice produced? If we could magnify 
the records cut by a phonograph in the wax cylinder, what would 
their shapes be? How are the possible successions of chords 
in a musical composition dependent on the tone quality and the 
beats. These inquiries can' not be pursued here, for a dis- 
cussion of them would lead us far beyond the scope of this book. 


1. Tone quality is related to wave shape. 

2. The addition of simple sound waves produces waves of 
complex shape. 

3. A single vibrating body may send out complex waves. 

4. The complex vibrations of strings and air columns are 
composed of simple vibrations whose numbers are related by the 
ratios 1: 2: 3, etc. 

5. The lowest note in the complex tone is called the funda- 
mental and the others are overtones. 

6. The relations between the vibration numbers of the 
fundamental and the overtones of strings and air columns are 
expressed by the ratios 1: 2: 3. 

7. Tone quality depends on the number, pitches, and rela- 
tive intensities of the overtones. 


8. An overtone in a compound note may produce resonance 
just as if it were alone. 

9. Two tones are discordant when either the fundamental or 
any of the overtones combine to produce disagreeable beats. 

10. The notes of the musical scale must be related by simple 
ratios because the overtones of musical instruments are so related. 

11. Harmony depends on tone quality as well as on pitch. 


1. Why may we assume that tone quality and wave shape are 

2. How do we prove that this assumption is correct? 

3. How can a single vibrating body send out complex waves? 

4. How do we know that the numbers of vibrations of the over- 
tones of a string are related by the simple ratios 1: 2: 3, etc.? 

5. Upon what does the tone quality depend? 

6. Can one component in a complex wave act to produce resonance 
in a body whose number of vibrations agrees with its own? 

7. When the fundamentals do not produce disagreeable beats, 
why may two tones still be discordant? 

8. What do we mean when we say two tones are musically related? 

9. Why do the notes of the musical scale have to be related by 
simple ratios because the overtones of strings and air columns are so? 

10m Could we replace" the strings of a piano with bells with good 
musical effect? If not, why not? 


1. Beginning with c = 24, write out the vibration numbers of 
the first 8 overtones of a string of that pitch. Do the same with the 
tone g = 36. How many of the eight overtones are common to both 
tones? Which of the overtones is the first common one? Write out 
the first eight overtones beginning on / = 32, and also on e = 30. How 
many overtones has each of these tones in common with the series 
beginning on 24? Which of the overtones in each series is the first 
common overtone? Which is the best consonance, c-g, or c-e? Which 
pair have the greatest number of common overtones?, 

2. Have the two tones beginning respectively on c = 24 and d = 
27 any of their first eight overtones in common? Do any of their over- 
tones give disagreeable beats, i.e!, more tlian four and less than 30 per 
sec? Is this interval c-d more or less consonant than the interval c-g? 
Can you see any connection between the consonance of a musical interval 


and the number of the first overtone wliicli is common to the two 
component notes? 

3. In a string on a musical instrument a node cannot exist at the 
point where the string is either bowed, picked, or struck with a ham- 
mer. If a string is plucked at a point distant from the bridge J the 
length of the string, what overtones will be wanting in the tone pro- 

4. A piano hammer strikes at a distance of } the length of the 
string from one of its ends. What overtones are wanting in the tone 
produced? Write the series of overtones for c = 256, and see if the 
seventh is apt to cause beats with its neighbors. Can the quality 
of the tone of a piano string be varied by changing the position of 
point where the hammer strikes? Why does a violinist bow near the 
bridge when he wishes to produce "brilliant" tones? 


1. Construct a vibrating flame as described in Art. 338, in Hop- 
kins's Experimental Science j and in Mayer's Soundj and see what vowel 
sound gives the most interesting vibrations. See if each of your voices 
gives the same shaped flame when singing the same vowel on the 
same pitch. Try other musical instruments in the same way. See 
if you can photograph the flame. 

2. Can you find out how a phonograph or a graphophone works? 
What sort of curves must be cut in the cylinder or disc of the machine? 
Have you ever examined such a curve with a microscope? 

3. Ask the organist at your church how organ pipes are made to 
have different qualities of tone. Examine the pipes yourself and see 
if you can think why the diameters of the flute and violin pipes are 
smaller in proportion to their lengths than those of the diapason pipes. 

4. Pronounce the vowels. In which is the mouth cavity elongated 
so as to give resonance to the lower overtones? In which is it short- 
ened so as to cut them out? 


347. What Does Light Do for TJs ? A peculiar interest at- 
taches to the study of light, because of its great usefulness to 
mankind. Not only is it indispensable for all human action, 
but also the color combinations by which it enables us to express 
art ideals are sources of highest pleasure and satisfaction. Can 
you conceive of a world devoid of light? And what a monotonous 
existence we should lead if light were deprived of color I Yet 
the very omnipresence of light often leads us to overlook its vast 
importance to life in the universe. In taking up the discussion 
of this subject, then, let us ask, first, what does light do for us? 

When we ponder this question carefully, we are led to con- 
clude that light enables us to gain information of four different 
kinds. 'First, it enables us to distinguish differences in the 
directions in which objects are located with reference to us and 
to one another. This ability to recognize differences in direction 
enables us to determine the shapes of objects as well as their 
relative positions, for the different parts of an extended object 
lie in different directions from our eyes. 

In the second place, light makes it possible to distinguish 
between the colors of things. This power not only assists us in 
distinguishing between objects about us, but it also enables us, 
as we shall presently see, to observe the peculiarities of distant 
stars and study the mechanism of ultimate atoms. 

In the third place, we are able to distinguish between intense 
and faint light — to recognize all the possible gradations of light 
and shade, whose totality produces the pictures which succeed 
one another with endless variety and which produce in us emo- 
tions of joy or pain, of inspiration or dejection, throughout 
our entire conscious lives. 

And, lastly, we can not only appreciate simple color, but we can 



distinguish and produce endless shades and varieties of color by 
mixing the simpler colors together in different ways. This last 
power, which we could not have without light, is fundamental in 
the art of painting, and is thus of far-reaching importance in 
our appreciation of the beautiful both in nature and in art. 

Now, it may seem to many sacreligious to attempt to pry into 
the mechanism of light — to seek to find out how light is able to 
do all this for us. We must confess that we think that it would 
be so if it were not for the fact that this inquiry does not in any 
way destroy our recognition of the enormous utility of light nor 
diminish our appreciation of the beauties of nature and of art 
which it enables us to enjoy. On the contrary, the detailed 
study of the phenomena of light, in the way in which the physicist 
studies it, adds enormously to our estimation of the wonders of 
light by showing us the ingenious way in which it operates to 
serve us as it does. 

348. What is the Nature of Light? After noting carefully 
the common experiences with light, the first t*hing the physicist 
does, is to ask what assumption or hypothesis he can adopt that 
will enable him to group the various phenomena together and to 
construct a mechanical model that will assist him in describing 
its action more in detail. When asked to propose such a hy- 
pothesis, what shall we say? We have just analyzed the kinds 
of information that light helps us in acquiring, and find them 
to be of four sorts, viz.: 1, As to direction; 2, as to color; 3, as 
to intensity; and 4, as to blending of colors. What sort of mech- 
anism is able to bring us such information? Probably, just as 
in the case of sound, a wave motion would suffice; and therefore 
we will at the outset adopt the hypothesis that light is a wave 

But what characteristics of the phenomena of light may we 
identify with each of the characteristics of the wave? Cleariy, 
the sense of direction is derived from the direction of propagation 
of the waves. It is also clear that the inttmsity of the light cor- 
responds to the amplitude of the waves. This leaves perception 
of color and the composition of colors to correspond respectively 

LIGHT 371 

to the wave length and the wave form. Possibly simple color 
may correspond to wave length and complex color to wave 
form. Let us, then, assume that these are the relations and see 
to what conclusions we shall be led. To this end we must enter 
upon a more detailed discussion of these characteristics of light. 

349. Direction. The first question that naturally arises 
is, how do we detect differences in direction? You answer, 
"With our eyes"; but how do they operate to detect the direction 
of propagation of waves? In order to answer this question, we 
must first call to mind a very familiar characteristic of light, viz., 
that it appears to travel in straight lines. Thus, the sunlight 
falling on the floor traces an outline of the window there. If we 
cover the window with a shutter having a small hole in it, we. 
notice that the beam of light which passes through the hole, and 
whose path is revealed by the dust particles in the air, travels 
in a straight line, and makes a bright spot on the floor. We also 
notice that the path of the light is the continuation of the line 
joining the sun and the hole in the shutter, so that if we invert 
the process and draw a line from the spot on the floor to the hole, 
that line indicates the direction of the sun. Thus we see that 
we can determine the direction of the sun with reference to the 
shutter and the floor, because the light travels ordinarily in a 
straight line. 

350. Image. If now we have two bright objects outside the 
window, like two electric lights, each will produce a bright spot 
on the floor or on some suitable screen held behind the hole in 
the shutter. When we draw straight lines from these two spots 
to the hole, they inclose an angle between them, and by means 
of this angle we are able to judge of the relative positions of 
the two electric lights. If now we have a large number of such 
bright points, for example a landscape outside the window, each 
point of the landscape produces on the screen a bright spot, which 
indicates the direction of the point with reference to the screen 
and the hole; these bright spots on the screen will each indicate 
the direction of its corresponding source, and so we obtain on the 
screen an image of the landscape outside (Fig. 198). 



Two things are apparent concerning the image thus formed: 
1. It is inverted, and 2, it is indistinct* and fuzzy. A moment s 

FiQ. 198. The Image is Inverted and Fuzzy 

thought will show us why it is inverted, namely, because the rays 
all cross at the hole, so those that were below on one side are now 
above on the other, and vice versa. Therefore this characteristic 
of the image is inherent in the nature of the phenomenon and 
can not be altered. 

But why is the image fu^zy? An analysis will show us why. 
It is because each point of the land- 
scape is sending out waves which 
spread out in all directions about that 
point (c/. Fig. 172). When these waves 
reach the hole in the shutter, they are 
divergent, and therefore make on the 
screen a spot of light somewhat larger 
than the hole, as shown in Fig. 199. 
Thus the image of each point of the 
object is a spot of light on the screeen, 
not a point; and therefore the entire 
image, which is the sum of these 
spots, is not clear and distinct like the object. Yet, even so, the 
image is enough like the object to be readily recognizable. 

Fio. 199. The Spot is Larger 


LIGHT 373 

351. What the Eye Does. Now, although this image is not 
as clear as the landscape outside, it enables us to distinguish 
definitely the directions of the different points of the latter. We 
might, therefore, conceive that the eye is able to distinguish 
directions by a similar device; for is not the pupil of the eye 
merely a small hole in a shutter, and therefore there must be 
formed at the back of the eye an image of the object in front? 
Now, an examination of the construction of the eye shows that 
immediately behind the pupil there is a little transparent object 
of hard elastic substance, called the crystalline lens (L, Fig. 
200). The front and back of this lens seem to be portions of 

Fig. 200. The Image Formed in the Eye 

spherical surfaces, and it is thicker at the middle than at the 
edges. Behind this lens the eye is filled with water, and the rear 
surface, called the retina, is covered with fine nerve filaments. 

352. What a Lens Does. We can see that an image would 
be formed at the back of the eye without the crystalline lens. 
What, then, is the use of this addition? On holding a piece of 
glass that is shaped like the lens of the eye behind the hole in the 
shutter, we observe that when the screen is at one particular 
distance from' the hole the image of the landscape outside is very 
clear and distinct. Therefore we may conclude that the purpose 
of the crystalline lens in the eye is to render the image on the 
retina distinct, i.e., to bring the rays from a point on the object 
outside together on the retina in a point instead of in a spot. Re- 
ferring to Fig. 204, p. 377, we see that such a lens must be able to 



bend the light, so that, after passing the lens, it is convergent 
instead of parallel or divergent. 

Fig. 201. The Light is Bent 
WHEN IT Enters the Water 

353. How Light is Changed in Direction. But if light 
travels in a straight line, how can a lens bend it? Yet, clearly, it 

does do so. Have you ever noticed 
that Ught bends when it passes 
obliquely from air into water? Place 
a battery jar full of water so that 
the sunbeam from the hole in the 
shutter falls obliquely on the surface, 
Fig. 201. Does the path of the light 
have the same direction in the water 
as it does in the air? Is [the path 
in the water straight? Thus it be- 
comes clear that light travels in a 
straight line only so long as it is 
moving through the same sort of matter, for when it passes from 
air to water it is bent. A similar effect is observed when we 
pass the light into glass or into any other transparent substance. 
How may we conceive that this bending is effected? We have 
assiumed that light is a wave 
motion. Let us then imagine 
that we have a beam of light 
of width a b, Fig. 202, traveling 
in air and approaching a sur- 
face of water a c. Let the direc- 
tion in which the light is trav- 
eling be represented by b c. 
Then the front of the wave, i.e., 
the line joining those points of 
the wave that are in the same 
phase, will be represented by 

a by which is perpendicular to b c. When the light has entered the 
water, we find that it is traveling in the direction c e; so that the 
wave front, which in the water is perpendicular io ce the new direc- 
tion of travel, has been turned from the direction a 6 to that of c d. 

Fig. 202. Refraction 

LIGHT 375 

This result would be accomplished if that portion of the wave near b 
traveled the distance 6 c in air in the same time that was taken by 
the portion of the wave near a to travel the distance a d in water. 
But ad \s clearly less than b c. So we see that we can form a 
perfectly intelligible picture of the manner in which a ray is bent 
on passing from one medium to another, by assuming that the 
light waves travel more slowly in the water than in the air. 

354. Index of Eefrax^tion. This phenomenon of the bending 
of a beam of light when it passes from one mediium to another 
is called refraction. How can we measure it? Clearly, the 
amount of bending depends on how much difference there is be- 
tween the velocity of light in the two media. For if 6 c remains 
the same, then the less ad is the greater the bending will be. 
We may, therefore, measure the bending by the ratio of 6 c to 
a d. But a d represents the distance traveled by the light in the 
second medium in a certain time t, and b c represents the distance 
traveled in the first medium in the same time t Therefore ad 
and b c are proportional to the velocities of light in the two media. 
So we may say that the amount of bending depends on the 
ratio of the velocities of light in the two media. Since we can 
measure the amount of bending by the ratio of these two veloci- 
ties, we call that ratio the index of refraction. It is usually 
denoted by n. Therefore we define this index in the following 

_ velocity in first medium _ b c , 

velocity in second medium ad' 

It has been proved by experiment that the velocity of light in a 
given medium is constant for a given color, therefore we may 
infer that this index remains constant for the same two media. 
It is not always easy, however, to measure the velocities in 
the two media; therefore let us see if there is not a more 
convenient form of expressing this ratio. To do this, drop a 
perpendicular n(m/ to the surface between the two media (Fig. 
203). Then the angle neb, formed between this perpendicular 
and the direction in which the light is traveling in air, is called 



the ANGLE OF INCIDENCE and is usually denoted by i. Similarly, 
the angle nfce is called the angle of refraction and is usually 
denoted by r. Now, from the figure we see that ^ neb = ^ bac = i 
and ^nfce = ^acd = r. We have also learned (c/. Art. 304) that 

itin bac = — = 9in i, and sin acd = — = sin r, whence 
ac ac 

sin I 



sin I 

^^ = ^ = ^. Therefore [ef. equation (18)] n^ ^^ (19). 
sin r ad ad ^ sinr ^ 


Thus tlw index of refraction is -equal to the sine of the angle of 
incidence divided by the sine of the angle of refraction. Since 

these angles are easily measured, this 
ratio is more convenient to use than 
that of the velocities. 

Now, if we use light of one color 
and measure the angles r that cor- 
respond to various angles of incidence 
i, and then, with the help of a table 
of sines, find the corresponding values 
of 71, we shall find that the values of 
Fig. 203. Refraction Diagram »i are the same for all angles of inci- 
dence. Therefore we conclude that 
n is constant for any two media and for any definite color ^ as 
we have inferred that it should be. . This fact was discovered 
experimentally by Willibrode Snell in 1680, and is known as 
snell's law, or the law of refraction. 

356. How the Lens Forms the Image. We are now able 
to see why the introduction of a lens of the form of the crystalline 
lens of the eye improves our image; for since that lens is thicker 
in the middle than it is at the edges, it is evident that those por- 
tions of the wave which pass through the center of the lens travel 
through a greater thickness of glass. But the light travels more 
slowly in glass than in air, so the center of the wave is retarded 
more than the portions near the edges of the hole, and the 
wave is converted from a plane or a convex wave into a concave 



■> ^ ■> 

5 -o 

FiQ. 204. The Light is Brought to 
A Point 

wave, as shown in Fig. 204, in which the vertical lines represent 
the successive wave fronts. 

Because . after passing the lens the wave fronts are concave, 
they contract toward a point 0, and there form a small image 
of the point from which they 
started. Thiis we see that 
introducing a lens of the 
given form contracts the 
spot of light to a point. 
Therefore every luminous 
point of the landscape is 
represented by a single 

bright point on the screen, and so the image becomes brilliant 
and distinct. 

We see, however, that the screen must be placed at the definite 
distance from the lens in order to receive a distinct image. 
This distance from the lens to is called the focal length, and 
the point is called the focus. Manifestly, the focal length will 
depend on the curvature of the lens, its index of refraction, and 
the shape of the incident wave. The study of the relations be- 
tween these quantities is of great importance, for the construction 
of all optical instruments depends on them. We can not properly 
understand optical instruments without a clear conception of these 
relations. We shall take up this study in the next chapter, as 

our attention is first demanded by one 
other important phenomenon con- 
nected with our perception of the 
direction of light. 

Fig. 205. Reflection 

356. Reflection. We can best 
study this phenomenon by placing a 
mirror on the floor where our sun- 
beam falls. The beam is turned^ 
away from the floor and reflected to the ceiling or to some 
other part of the room. Fig. 205. If we turn the mirror into 
different positions, we note that the reflected beam is turned 
in different directions. But by so turning the mirror we vary 



the angle at which the incident beam falls upon it. We also 
vary the angle at which the reflected beam leaves it. Is there 
any relation between these two angles? 

!Fi6. 206. Reflection Diagram 

367. Laws of Eeflection. Before answering this question, we 
must agree as to how we shall measure the angle of incidence. We 

therefore define the angle of inci- 
dence to be the angle included be- 
tween the perpendicular to the sur- 
face and the incident beam. This 
angle is measured in the plane 
which contains this perpendicular 
and the incident beam. Thus if 
(Fig. 206) AB represents the direc- 
tion of the incident beam, BC that 
of the reflected beam, and NB the perpendicular to the mirror, the 
angle of incidence is then NBA, This angle is, as before, denoted 
by i. Similarly, the angle of reflection is that included between the 
perpendicular NB and the reflected beam 5C, i.e., it is NBC. 

We may now ask what relation, if any, exists between these 
angles. In order to answer the ques- 
tion, we must measure various angles 
of incidence and the corresponding 
angles of reflection. If w^e do this, 
we find that tJie angle of incidence is 
equal to the angle of reflection in all 
cases. We further find that the in- 
cident beam, the perpendicular, and 
the reflected beam, all lie in tJie same 
plane. Therefore we conclude that these are the laws of re- 

Fig. 207. The Plane Mirror 

358. Where the Image Appears. These principles can now 
be used to find out where a source of light aj)pears to be when we 
observe it by reflection in a mirror. Clearly, an observer at C, 
Fig. 206, will receive the light as if the source were in the direction 
CB, But at what point in that direction? In order to find out, 

LIGHT 379 

we have but to alter Fig. 206 as follows: Let S (Fig. 207) be the 
source of light. Since it is sending beams in all directions, it 
will send out not only one in the direction S B, but also many 
in other directions. One particular ray SD will strike the 
mirror perpendicularly, and be reflected back along its own path, 
i.e., in the direction of DS. 

Now, an observer at C sees the light in the direction CB and 
another observer behind S and in the direction DS sees it in the 
direction SD. In what direction does the source appear to lie? 
Clearly in the directions of both lines. Hence the apparent source 
of the light must be at the point S', where the lines CB and SD, 
intersect when they are extended behind the mirror. 

But where is the point S' with respect to the mirror? From 
the law of reflection, we know that the angle NBS == angle NBC, 
hence we may easily prove that the two right triangles SBD and 
S^BD are equal, and therefore S'D = SD, But these are the 
distances of the source S and its image S', measured perpendicu- 
larly from the mirror MM; so we see that this result may be stated 
as follows: When light is reflected in a plane mirror the image 
appears to he as far behind the mirror as the source is in front of it. 

359. Diffuse Eeflection. The law of reflection just stated 
applies clearly to all cases of reflection from metallic or other 
polished surfaces. If we replace the mirror by a piece of white 
paper, what becomes of the reflected beam? If we try the ex- 
periment with other things by placing them in the path of the 
sunbeam we will notice that some of them act like the mirror and 
reflect most of the light in a definite beam while others reflect 
part of it in a definite beam, and still others reflect none of it in a 
definite beam. Hence we see that this law of reflection is not 
general in its application. We may, then, ask what law applies 
to these other cases. On placing a white card in the path of the 
beam, we note that the light seems to be scattered in all directions 
from the bright spot on the card. In fact, the effect is the same 
as if the card were itself the source of the light. This phenom- 
enon is called diffuse reflection in contradistinction to the 
other kind, which is called metallic reflection. 


The importance of diffuse reflection is seldom appreciated. 
We do not often realize that we see most objects because they 
reflect diffusely. Thus, when we look at a landscape or a picture, 
each part of the object affects us as if it were itself a source of 
light. This, then, is the law of diffuse reflection, viz.: A body 
that reflects light diffusely appears as if it were self-luminous. 

We thus see that there are two kinds of reflection, metallic 
and diffuse, of which the latter is the more important to mankind. 
It is not possible, however, to classify all substances as reflecting 
either metallically or diffusely. At one end of the series we have 
the metals, which reflect in the first way only; and at the other 
end we have what are called perfectly matt surfaces, like a plas- 
ter wall, which reflect entirely diffusely. Between these two extremes 
we have substances that reflect partly in one way, partly in the 
other, in all sorts of varying proportions. 


1. Light enables us to acquire four kinds of information: 
1. As to direction of a source; 2, as to its color; 3, as to its in- 
tensity, and 4, as to its tone of color. 

2. Waves also bring us four similar kinds of information, and 
therefore we adopt the hypothesis that light is a wave motion. 

3. Light ordinarily travels in straight lines in any one medium. 

4. An image of an object is formed when the light from it 
passes through a small hole. 

5. Such an image is inverted and blurred. 

6. A lens makes this image distinct and more brilliant. 

7. The focal length of a lens is the distance from the lens to 
the point at which the image is formed. 

8. The conditions necessary for the formation of a clear image 
are realized in the human eye. 

9. The direction in which light travels is altered when it 
passes obliquely from one medium to another. 

10. The amount of this bending is measured by the index of 

IL The index of refraction is the ratio of the velocities of 

LIGHT 381 

light in the two media, or the ratio of the sines of the angles of 
incidence and refraction. 

12. The index of refraction is a constant for any two given 
media and for a given color. 

13. When light is reflected from a metallic surface, the angle 
of incidence is equal to the angle of reflection, and lies in the 
same plane. 

14. The image of an object reflected in a plane mirror appears 
as far behind the mirror as the object is in front of it. 

15. Light is diffusely reflected from unpolished surfaces. 

16. A surface reflecting diffusely appears as if it were itself 
a source of light. 

17. The reflection of light by many surfaces is partly diffuse 
and partly metallic. 


1. What four kinds of information does light enable us to acquire? 

2. What may we assume as a working liypothesis as to the nature 
of light? 

3. Upon what property of light does our determination of the 
direction of light depend? 

4. How is an image formed through a small hole? 

5. Why does a lens improve the clearness of such an image? 

6. Describe the construction of the human eye. Wliat provision 
is there made for distinguishing differences in the directions of objects? 

7. Why is the path of light bent when it passes obliquely from air 
into water? 

8. How do we measure the amount of this bending? 

9. What relation exists between the index of refraction and the 
velocities of light in the two media? between that index and the angles 
of incidence and refraction? 

10. What is the focal length of a lens? 

LI. What is the difference between metallic and diffuse reflection? 

12. What is the law of diffuse reflection? 

13. Which kind of reflection is the more common? Which is the 
more aseful? 

14. Where does an object reflected in a plane mirror appear to be? 
Can you prove it by the geometrical relations? 

15. What sorts of substances reflect metallically? What sorts 
entirely diffusely? 



1. The method of finding the location of the image of a point source 
in a plane mirror is described in Art. 358. Replace the point source 
by an arrow and graphically construct the image. How far behind 
the mirror does the image lie? 

2. If Vi represents the velocity of light in glass and V2 that in water, 
show that the index of refraction of light at a surface between glass 

and water is equal to — . If v represents the velocity in air, what is 

the index at the surface between air and glass? Between air and 
water? May the index for glass-water be obtained by dividing that 
for air-glass by that for air- water? Show how. 

3. The index of refraction is shown to be the ratio of the sine 
of the angle of incidence to the sine of the angle of refraction, Art. 354. 
If the light falls perpendicularly on a surface of glass, so that the angle 
of incidence is 0, what is the value of the angle of refraction? Is the 
light bent when it passes perpendicularly through the dividing sur- 

4. Is the period of vibration of a light wave (i.e., the color of the 
light) changed by passing from one medium to another, as from air to 
water? With the help of the equation v = n Z, show that the index of 
refraction n^ay also be defined as the ratio of the wave length of the 
light in air to its wave length in water. 

5. When light passes from water into air, is the path of the light 
bent toward or away from the perpendicular to the surface? How 
do you define the index of refraction under these conditions if Vj rep- 
resent the velocity of light. in water and v its velocity in air? How 
is it defined in terms of the sines of the angles of incidence and refrac- 
tion? How does its numerical value compare with that for the con- 
verse case of light passing from air into water? 

6. The index of refraction of air- water has the value 1.33, i.e., 

— = 1.33, or sine i = 1.33 sine r, But'the sine of an angle is the 


ratio in a right triangle of the side opposite the angle to the hypoth- 

enuse, and since that side cannot be greater than the hypothenuse, 

sine i cannot be greater than 1. What is the greatest value that sine r 

can have? 

7. If we reverse the direction of the light and send it from water 
into air, is there any reason why the angle r should not be greater 
than the value determined as a maximum in problem 7? If we give 
the beam a greater inclination to the surface, so that the value of 1.33 
sine r becomes greater than 1 , what becomes of sine i? What does 
the light do? Can it escape from the water? This phenomenon is 
called total reflection. 

LIGHT 383 


1. Using the principle that the angle of incidence is equal to that 
of reflection, see if you can find out by graphical construction what 
will become of a beam of parallel light after it is reflected from a con- 
cave -spherical mirror. If you have a spectacle, camera, or opera glass 
lens that has a concave surface, reflect the sunlight from it and see 
if your construction is correct. Try the same experiment with a silver 
spoon or a lamp reflector. Can you construct graphically in the same 
way the image of an arrow as formed by a concave mirror? 

2. Make a pin hole camera and see if you can take a picture with it. 
Any light-tight box will do. To get fairly clear images, the edges of 
the pin hole must be smooth. 

3. Reflect a sunbeam into a neighboring window with a plane 
mirror. When you turn the mirror through any given angle, through 
what angle is the reflected beam turned? See if you can devise a method 
of measuring it with a protractor and of making a graphical solution. 
The geometry of the right triangle will aid you here. 

4. How are search lights constructed? What is the shape of the 

5. Were you ever in a "crystal maze?" If so, explain your per- 
plexities in a brief paper. 

6. Why is a large mirror of advantage in decorating a small room? 
Would a living room be comfortable if its walls were nearly covered 
with metallically reflecting substances? Can you see a physical reason 
why matt surfaces are considered in better taste? 

7. Make a diagram to prove that the lower half of a full length 
mirror is not necessary in order that a lady may see her entire figure 
in it. Verify the conclusion by experiment. 

8. Find out why most books are printed on matt paper. 



360. Principal Focus. In the last chapter we learned that 
an image of a luminous point is formed by a lens at a particular 
distance from the lens. Is this distance always the same for a 
given lens, no matter where the luminous point is situated with 
reference to it? In order to answer this question, take a simple 
lens L of the shape shown in Fig. 208, and allow light from the sun 
to fall on it in the manner there shown. On holding a paper 
behind the lens, we easily find the point at which the image of 
the sun is distinctly formed. Now, the 
sun is so far away that the wave fronts 
of the waves that reach us are sensibly 
" plane. Hence they may be represented by 
straight lines as AB, which are moving in 
a direction LF perpendicular to AB. Since 
BD AB \s a. straight line, all the perpendicu- 

FiG. 208. Principal lars to it, which indicate the directions of 
motion of the various parts of the wave, are 
parallel to one another and to LF. These lines are called rays. 
Such a series of parallel rays constitute a parallel beam. In 
order to make the figure symmetrical, let us place the lens so that 
its central plane CD is also perpendicular to LF, The line LF, 
which passes through the center of the lens and is perpendicular 
to the plane CD, is called the optical axis of the lens. 

Since the light waves in this case constitute a parallel beam, 
and since they are moving in the direction of the axis of the lens, 
it is evident from the symmetry of the figure that they will be 
brought together at a point on that axis such as F. Then F 
will be the focus. We note that the direction of motion of the 
central part of the wave has not been changed while the outside 
portions have been bent through lACF. The point F, at 



which parallel rays are brought together, is called the principal 
FOCUS. In the case under consideration, we may measure the 
distance ZF, and this is the principal focal length. 

361. Image of a Point Source. If, with our lens, we form an 

image of some object that is not very distant, say of a luminous 

point S on the 

axis. Fig. 209, the 

image / of that , 

point will lie 

farther away from 

the lens L than ^ , ^ « 

. , Fig. 209. Imagk of a Point Source 

the principal focus 

F, because the incident waves from a near point are not plane but 

convex, and, since the thickness of the lens is the same as before, 

and the light passes through its center in the same direction as 

before, the same retardation will be produced in the center of the 

beam. But part of that retardation is now necessary to render 

the incident waves plane, and so less is left to make the plane 

waves concave; therefore, they come to a focus at sonae point I 

farther away from the lens than the principal focus F, 

If we bring the point S still nearer to the lens, we find that its 

image is still farther away (Fig. 210), and when the distance of 

the point S from 

• the center of the 

I lens is equal to 

the principal focal 

length, the waves 

behind the lens 

Fig. 210. As the Source Approaches the Image become plane and 


we have a parallel 
beam, Fig. 211. We see that this figure is the converse of Fig. 208. 
This reciprocal relation is general in optics. // the source is placed 
where the image was, the image will he found where the source was. 

362. Characteristic Rays. In discussing the formation of 
images of extended objects by lenses, it is simpler to consider 



the rays only, and not the waves; so we shall use rays in the re- 
mainder of the explanation. 

In the general discussion just given, we note that in the forma- 
tion of an image two rays are particularly well defined, viz., the 
one that passes through the center of the 
lens, and the one that is parallel to the 
axis. Since the ray that passes through 
the center of the lens is not bent, its path 
is determined by the point source and the 
center of the lens. Since rays parallel to 
the axis, after passing the lens, go through 
the principal focus, this ray is determined" 
by the distance of the ray from the axis, 
the direction of the axis, and the principal focus. These two 
rays enable us to find out many important things concerning the 
relations between objects and images formed by lenses. 

Fio. 211. Source at the 
Principal Focus 

363. Construction of the Image. For example, suppose 
that we have an object 00', Fig. 212, at a distance ML in front of 
the lens whose principal focus is at F, Where ^ill the image 
of the object be? Its position may be found as follows: From 
O draw a ray through the center of the lens L. This ray passes 
through the lens without being deflected, and the image of O must 
lie on this line at some point, as /. Similarly, a ray from 0' through 
the center of the lens passes through the lens without being de- 
flected, and the image of 0' must lie on this line at some point, as 
r. Thus it appears 
that the image must 
lie within the angle 
ILP. We shall call 
this angle the lens 
ANGLE of the image. 
The angle OZO' will 
be called the lens angle of the object, and we see that the two are 
equal. The lens angle will be found to be very useful in discuss- 
ing optical instruments. It is defined as the angle subtended at 
the center of the lens by either the object or the image. It will 

Fig. 212. Construction for the Image 


be noted that the lens angle depends only on the size of the object 
and its distance from the lens, and is independent of the size or 
shape of the lens used. 

In order to locate the points I and'/' on the sides of the lens 
angle, from draw a ray parallel to the axis MF. This ray 
must pass through the principal focus, and the image of must 
lie on it. Hence the image of must lie at the intersection / of 
the two rays FI and LI, When we have located the image /' of 
the point 0' in the same way, it will be noted that the image is in- 
verted, as was found to be the case, in the last chapter (Art. 350). 
This simple construction is very useful and always gives us a close 
approximation to the position of the image; for both the center of 
the lens and the principal focal length can always be determined 
with a fair degree of accuracy, the former from the geometrical 
symmetry of the lens, and the latter by fonning an image of the 
sun, as just described. 

364. Size and Distance of the Image. What are the relative 
sizes of the objects and the image? To understand the answer 

to this question, we 
must first distinguish 
between angular and 
.LINEAR size. If we 
^i mean the former, the 

Fia. 213. Relative Sizes op Object and Image ^^^^ ^* ^^^ image IS 

the same as the size of 
the object, i.e., the angle subtended hj the object at the center of a 
lens is tlie same as that subtended by the image at the same pmnt 
This fact must be carefully noted, because it is fundamental in 
understanding the operation of optical instruments. 

The relative linear sizes of the object and its image are differ- 
ent, however, for they are at different distances from the lens. 
In Fig. 213 the object OM and the image IN are both perpen- 
dicular to the axis MN of the lens and their lens angles are equal, 
i.e., /.OLM = /.ILN, Therefore the right triangles ILN and 

OLM are similar, and j%j ^ y?J' ^^' ^^ linear size of the object 


w to the linear size of the image as the distance of the object from 
the lens is to the distance of the image. These two distances arc 
therefore of great importance in the discussion of relative linear 
sizes. They are called the conjugate focal lengths. Simi- 
larly, the points M and N, so related that an object at either of 
them forms an image at the other, are called conjugate foci. 

We can now determine the relative linear sizes of object and 
image. We see that when M L is greater than L N the object is 
larger than the image; i.e., when the object is farther from the lens 
than the image, the object is larger. Conversely, when the iwxige 
is farther from the lens than the object, the image is larger, and 
when object and image are at equal distances from the lens they 
have the same size. In this latter case the distance of the object 

or the image from the lens is twice 
the principal focal length of the 
lens. In this position the object 
and the image are as hear to- 
gether as they can be. Thus we 
see that as an object is brought 

Fig. 214. The Object is at the au i xu • 

Pkincipal Focus nearer the lens, the image re- 

cedes from the lens and becomes 
bigger as every photographer knows. This increase in the size 
of the image is due both to an increase in the lens angle as the 
object approaches, and to the increase in the distance of the 
image from the lens. Is there any limit to this process? Can 
we bring the object close to the lens and get an infinitely large 
image? Let us see. 

We have already remarked that when the distance of the 
object from the lenii is equal to the principal focal length of the 
lens the emergent rays are parallel. Do such rays meet? Where, 
then, is the image formed? We may conceive that an infinitely 
large and infinitely distant image is formed when the distance of 
the object from the lens is equal to the principal focal length. 
This case is illustrated in Fig. 214. 

365. A Virtual Image. Suppose we bring the object still 
nearer to the lens, where will the image lie? We may find out by 


constructing a diagram, Fig. 215. Evidently the rays are divergent 

after leaving the lens. Where, then, is their point of intersection 

at which the image is formed? They 

do not intersect after leaving the 

lens, so that they are miable to form 

an image on a screen. They do, 

however, proceed as if they came 

from a point behind the lens. Yet 

if we place a screen aj; that point, we 

have no image formed on it, for the 

rays do not actually intersect there. 

Hence, such an image is said to be 

VIRTUAL. It will be noted that this \ 

virtual image is not inverted, as real images are, and that in this 

case, also, the lens angle of the image and that of the object are 

the same. 

Fia. 215. TiiE Image is Virtual 

366. How the Eye is Focused. In Art. 361 we have learned 
that as the object is brought nearer the lens, the image is formed 
at a greater distance from it. In the eye, however, the distance 
between the lens and the retina, where the image is formed, is 
constant; yet we can see both distant and near objects clearly. 
Unless there were some means of focusing the image on the retina, 
it would not always appear sharply defined. The device employed 
is that of changing the thickness of the lens. For if (Fig. 209) the 
lens were made still thicker in the middle, the central portions of 
the beam would be retarded still more with respect to those at 
the rim, and the curvature of the waves would be changed more 
in passing through. This increased curvature would bring the 
waves to a focus nearer the lens. 

In order to bring the images of objects that are near-by to a 
focus on the retina, the rim of the crystalline lens of the eye is 
Surrounded with a small muscle which contracts, and squeezes 
the lens so that it bulges out and becomes thicker in the middle. 
Thus its focal length is shortened and the near-by objects brought 
to a focus on the retina instead of at a point behind it. The eye 
thus ACCOMMODATES itself to the different focal distances. This 




accommodation of the eye is limited. In a normal eye the limit 
is reached when the object is about 25 cm (= 10 inches) from the 

eye. If we bring the 
object nearer than this, 
the image recedes be- 
hind the retina, and 
since there is no dis- 
tinct image on the 
retina we can not see the object clearly. This distance — ^the least 
at which the object can he, placed from the eye and still form a 
distinct image on the retina — is, therefore, called the limit of 


Fio. 216. Far-Siohted Eye 

367. Spectacles. If the crystalline lens in the eye is not normal, 
it does not form clear images of objects at all distances down to 
25 cm. If it is too weak, i.e., too thin in the middle, the accom- 
modating muscle must be used in forming the images even of 
distant objects, and will have to squeeze the lens harder in order 
to focus near-by objects on the retina. Therefore this muscle 
rarely gets any rest while its possessor is awake. This defect of 
the eye is called far-sightedness (Fig. 216), and is likely to cause 
serious results unless corrected. Far-sightedness is corrected by 
strengthening the crystalline lens and making it thicker in the mid- 
dle. This necessitates introducing in front of the eye a lens which 
is thicker in the middle than at the' rim. This lens is indicated 
by ab in Fig. 216. The strength of the added lens must be 
such that distant objects are seen clearly when the accommodat- 
ing muscle is relaxed. 

Conversely, an eye is near-sighted when the crystalline lens 
is too thick in the mid- 
dle. Then the images of 
objects are distinct only 
when held very close to 
the eye, but those of 
distant objects are formed 
in front of the retina (Fig. 217). This defect is corrected by plac- 
ing in front of the eye a lens that is thinner in the middle than at 

Fig. 217. Near-Sighted Eye 



Fig. 218. Limit of Distinct Vision 

the rim, as indicated by cd in Fig. 217. In this case, also, 
the added lens should be of such curvature that distant objects 
are seen clearly when the accommodating muscle is relaxed. 

368. The Simple Microscope. We are now prepared to under- 
stand the operation of the simple microscope. This consists of a 
single lens which is thicker in the middle than at the rim. It 
therefore does not differ in its action from the far-sighted spec- 
tacle lens. It merely 
enables us to focus ^* 
clearly on an object 
which is much nearer 
the eye than the limit 
of distinct vision. Fig. 

218 shows the object at the limit of distinct 
vision, and Fig. 219 shows the same object 
brought nearer the eye and focused on the 
retina with the help of the microscope lens. 
It is clear that in this case the lens angle 
OLO' is greater than before. Since our ap- 
preciation of the size of an object depends on the angle that 
it subtends at the eye, and since that angle depends only on 
the size of the object and its distance from the eye, it is evident 
that the microscope enlarges the apparent size of objects, because 
it enables us to see an object clearly when it is very near the eye, 
so that its lens angle there is large. This lens angle of the eye 
is generally called the visual angle. 

369. The Camera. Besides the eye and the simple microscope, 
probably the camera is the best known optical instrument. We 
have just discussed (Art. 350) the formation of an image by a small 
hole in a shutter, and shown how that image is made clearer by 
the introduction of a convex lens. Every one must recognize the 
image formation thus produced as identical in every respect with 
that of the camera. But a camera lens consists not of a single 
reading glass, but of several lenses mounted together in a tube. 
Nevertheless, in this case also the lens angles of the object and ot 

Fig. 219. Simple 


the image are the same, since the combination of lenses may be 
replaced by a single lens that would produce a similar effect. The 
discussion of all the reasons for thus making the camera lens of 
several parts, would lead us far beyond the limits of our present 
study. The use of the stops, in such lenses, however, demands 

370. Stops. In the first place, every photographer knows that 
photographic lenses are supplied with stops which limit the amount 
of the lens used. The effect of a stop is twofold, viz.: 1, it reduces 
the amount of light admitted to the camera and so lengthens the 
necessary time of exposing the sensitive plate to the light that 
comes from the object through the lens; and 2, it makes the image 
on the plate sharper at the edges. The relation between the area 
of the opening in the stop and the time of exposure is simple; for 
the amount of light that enters the lens from a point on the object 
is proportional to that area, and therefore the intensity of the light 
at a point on the plate must vary as that area, other things remain- 
ing the same. Hence, with a given lens, a stop whose opening has 
half the diameter of the lens requires an exposure four times as 
long as that required for the lens without the stop, since the areas of 
the openings are proportional to the squares of the diameters. 
It will be noted that the introduction of the stop does not change 
the lens angle. 

371. Spherical Aberration. The other effect of the stop now 
demands our attention, viz., that the clearness of the image around 
the edges of the plate is improved by reducing the size of the 
opening in the stop. The reason for this is rather complex and 
requires for its complete explanation a more exhaustive study 
study than can be undertaken here. Suffice it to say, the theory 
shows that lenses whose surfaces are portions of a sphere can not 
bring all the points of a plane image to a focus in a plane. Thus, 
if we have a plane object perpendicular to the axis at M, Fig. 
220, the points of the object near the axis LN will be focused in 
a plane perpendicular to the axis at N. This plane is called 
the focal plane. But points of the object that are farther away 



FiQ. 220. 

The Image Does not Lie in a 

from the axis, will be focused either in front of or behind this 
focal plane. 

In order to bring all the points of the object to a focus in one 
plane, the surfaces of the lenses -would have to be portions of 
ellipsoids instead of 

spheres. But as ellip- l! ' " — — -..^.^^^^^ iL_-^-^ — ^^' 
soidal surfaces are al- 
most impossible to grind 
and polish, their use is 
practically out of the 
question. However, the difference between a spherical and 
an elliptical surface is small if we consider only a small area 
of each. Therefore, when we reduce the area of the hole in the 
stop, we allow only the central portion of the lens to be used, and, 
therefore, we make its difference from the theoretically correct 
shape very small. So the image becomes clearer. This 
blurring of the image because of the spherical shape of the lens 
surfaces is called spherical aberration. It has been found that 
the spherical aberration of a lens may be somewhat reduced by 
using several lenses instead of one, so that this is one reason why 
photographic lenses are made of several parts. Another reason 
will be discussed in the next chapter while studying color. 

372. The Astronomical Telescope. The next instrument to 
which we shall direct our attention is the astronomical telescope. 
This consists of at least two lenses of the type of the simple micro- 

FiG. 221. The Astronomical Telescope 

scope. Inasmuch as the telescope is usually used for observing 
distant objects, we shall assume that the beam of light from every 
point of the object is a parallel beam. Thus, in Fig. 221, let the 
line L I represent the path of the incident beam coming from the 


tip of a distant arrow and passing through the lens i, which is called 
the OBJECTIVE. If F is the principal focal length of L, the image 
of this point will then be at 7 in the principal focal plane. Simi- 
larly, the image of a point at the other end of the arrow will be 
at /'. The lens angle of the object is then equal to that of the image, 
i.e., it is the angle ILI\ If we introduce a second lens behind 
the image in such a way that its principal focus is also at F,> this 
second lens will render parallel the light from each point of IT, 
so that the beam from each point on the object is a parallel beam 
when it leaves this second lens as well as when is strikes the ob- 

The image IP is now the object for the combination consisting 
of the second lens and the eye, and its lens angle with respect to 
this combination is IL'I'. Since the image IV is nearer tp the 
eye combination than to the objective, its lens angle at that com- 
bination is larger than its lens angle at the objective, i.e., 
ILT > ILI'; and, therefore, the object appears enlarged. It 
will be noted that the image is inverted. Since the second 
lens is near the eye, it is called the eyepiece. The combination 
of an eyepiece or a simple microscope with the eye will be called 
the eye combination. 

The reason why the telescope makes things appear larger is 
now apparent. The visual angle of the object, when viewed 
without the telescope, is small^ because the object is usually at a 
great distance. This angle is very nearly the same as the lens 
angle of the object at the objective of the telescope. But the 
objective forms an image close to the eye combination. Since 
this image is close to the eye combination and at a much greater 
distance from the objective, its lens angle in that combination is 
larger than the visual angle of the object. The magnification may 
lens angle of the eye combination _ IL'T _^ lUF 
lens angle of the objective IhV ILF 

In Art. 6 we learned that an angle may be measured by its tangent, 

IF . . IF 

and that the tangent of IL'F = YTp- Similarly, tangent ILF = y-p- 

rpi . ,u -fi *• • tangent lUF LF 

Therefore the magnihcation is ^ — - — ; — ^^rir = -rm- 
^ tangent ILF LF 

Plate Vlll. The 40-inch Telescope, Yerkes Observatory, 

Copyright W05 hy the W r I T T AMS R A Y W m 

University of Chicago ^ ^ LLl A Mb 13 A \ , VY IS. 


But LF is the principal focal length of the objective, and L'F 
is that of the eyepiece; so we see that the magnification of the 
telescope is determined by the ratio of the focal length of the 
objective to that of the eyepiece. Therefore, if we wish to make 
telescopes that shall have large magnifying powers, we must so 
construct the objective that it will have a great principal focal 
length, while the eyejHece must be made to have a small focal 

Plate VIII is a photograph of the Yerkes telescope, which is 
the largest in the worid. It has a focal length of about 25 m; 
therefore, with an eyepiece which had a focal length of 0.5 cm, 
its magnification would be 5000. Such a high magnification can 
seldom be used, on accoimt of the unsteadiness of the atmosphere. 
Since the more the image is enlarged the fainter becomes the light 
in each cm* of it, it follows that when large magnificjitiohs are 
used large lenses are necessary, in order to gather as much light as 
possible into the image. The objective of this telescope is 100 
cm in diameter. 

373. The Concave Lens. If we examine a common opera glass, 
we find that the eyepiece is thinner in the middle than at the rim. 
Such a lens is called concave; those 
that are thicker in the middle being 
CONVEX. We have noted (Art. 367) the 
action of the concave lens in aiding 
near-sighted persons to see more clearly. 
To gain a better understanding of the 

.. « 1 'J i7« FiQ. 222. The Concave Lens 

action ot concave lenses, consider i^ig. Scattehs the Light 

222. Light waves from [the point S 

spread out and fall on the concave lens L, Since the lens is thicker 
at the rim than in the center, the waves that pass through the rim 
are more retarded than those passing through the center. The 
result is *that the divergent beams from S- are rendered more 
divergent, so that the waves behind the lens will appear to come 
from some point as I. Thus, / is the image of S, and it is virtual. 
When the point Sis faraway, so that the incident beam is parallel, 
the principal focus will be near / and will also be virtual. Con- 



versely, if we have a convergent beam that would otherwise come 
to a focus at the principal focus F of a concave lens, the interposi- 
tion of this lens will make the beam parallel (Fig. 223). If we place 
an arrow at S and construct the image as described in Art. 363, 
'"^^^^.^ we find that the lens angle of the ob- 

;; ^ ject is equal to that of the image, i.e.. 

OLO' = ILP, Fig. 213. 

Fig. 223. Parallel Beam 
Formed by CJoncave Lens 

374. The Opera Glass. Let us now 
construct a diagram to ^present the 
visual angles as they occur in the opera 
glass. The objective alone would form, an inverted image //' of 
the distant object at its principal focus F. The lens angle of the 
object is, then, equal to FLL The concave lens is introduced in 
front of the image, with its principal focus also at F. Then, since 
the rays which fall on i' from any point of the object are converging 
toward a point in the plane of its principal focus, they will be ren- 
dered parallel by the eyepiece (cf. Art. 373). The direction of each 
such parallel beam will be that of the corresponding point of the 
image //' from the center of the lens U, Hence, the lens angle 

Fig. 224. The Opera Glass 

of the image after passing the lens U will be IL'I; and there- 
fore, as in the case of the astronomical telescope, the magnification 
is the ratio of the focal lengths of the objective and the eyepiece. 
For viewing ordinary objects this instrument has an advantage 



over the telescope in that the image formed by it is upright. 
The opera glass is often called the Galilean telescope, since it 
is the kind invented by Galileo and with which he discovered the 
satellites of Jupiter. 

375. The Compound Microscope. This instrument differs from 
the telescope only in the fact that the object is placed in front of 
the objective near its principal focus, so that its lens angle 
may be made as large as possible. A real image is formed by 
the objective near the eye combination, so that the lens angle of 
this real image with respect to that combination is also large. 
Fig. 225 shows the arrangement, ILI' being the lens angle of the 

Fig. 225. The Compound Microscope 

object, and IW the lens angle of the image with respect to the 
eye combination. By using an objective of very short focal 
length, real iinages may be obtained when the object is brought 
very close to the objective. But, provided it remains outside of 
the principal focus, the nearer the object is brought to the ob- 
jective, the larger is the lens angle of the object, and also the 
larger i? the linear size of the image. Consequently, by this ar- 
rangement very large magnifications may be obtained, the only 
limit being the intensity of the light; for as the image gets larger 
it becomes fainter. When we reach a magnification of about 
2500 we have reached the practical limit for eye work. Photo- 
graphs can, of course, be made with still larger magnification3, 
but the exposures must be very long. 

One interesting point may yet be mentioned. Although we 
can magnify an object without limit, a point is eventually reached 


beyond which greater magnification fails to reveal further detail 
in the object.- What is that limit? Though we can not here give* 
the reasons for the conclusion, we may, nevertheless, state it. 
When the distance between two points of an object is less than 
1-100,000 of an inch, we are not able, by any known optical device, 
to distinguish whether there are two points or only one. Thus, 
when an object has been magnified imtil points one-hundred- 
thousandth of an inch apart are separated, further magnification 
will not reveal any further details of the construction of the object. 
This point is of interest, because the ultimate particles of matter — 
atoms and molecules — ^are much closer together than this in solids, 
and therefore we know that we can never see them with our 
eyes, though we may be able to know them in other ways. 

376. The Photometer. In the discussion of the astronomical 
telescope and of the microscope, we have found that the intensity 
of illumination of the image is a matter of importance. In prac- 
tical life it is a matter of even greater importance, since all artificial 
lighting by gas, or electricity, is measured and rated according to 
its intensity. The unit in which intensities are measured is the 
CANDLE-POWER. This is the rate at which light is radiated from 
a candle of specified construction burning a specified amount of 
sperm per minute. The ordinary electric glow lamps are generally 
equivalent to 16 standard candles, and are therefore called 16 
C.P. (candle-power) lamps. 

Intensities are compared by means of a photometer. The two 
lights to be compared, e.g., a lamp and a standard candle, are set 
about 2 or 3 m apart, and a piece of paper with a grease spot "on 
it is supported between them. This paper is moved backward or 
forward until the spot can no longer be seen. When this is the 
case, the illumination of the two sides of the paper is the same. 
The intensities of the two lights must be directly proportional to 
the squares of their distances from the paper. If the distance 
of the standard candle from the paper is 20 cm and of the lamp 
80 cm, or 4 times that of the candle, then the intensity of the lamp 
is 4* times that of the candle, or 16 C. P. This may be proved 
by considering that the light is spreading out in all directions 


from each source, so that the energy that spreads over 1 cm' 
on a surface 1 m from the light is spread over 4 cm^ on a surface 
twice as far from the light. This form of photometer is called the 
Bunsen photometer, after its inventor, and as we have just seen, 
it is based on the principle that the amount of light which falls 
from a given source on ea>ch cm* of a surface is inversely proportional 
to the square of the distance of the source from the surface. This is 
a geometrical consequence of the straight paths of the rays; but 
it is strictly true only when the distance is large compared with 
the size of the source, so that the light may be regarded as diverg- 
ing from a point. 


1. When parallel rays are brought to a focus by a convex lens, 
the distance from the lens to the focus is called the principal focal 

2. As the object is brought nearer the lens, the image recedes 
from it. When the image is real, it is inverted. 

3. When an image is formed by a convex lens, the ratio of 
the linear size of the object to that of the image is equal to th:) 
ratio of the distance of the object, from the lens, to that of the 
image, from the lens. 

4. When the distance of the object from a convex lens is less 
than the principal focal length, the image is erect and virtual. 

5. When an object and its real image have the same size, the 
distance of each from the lens is equal to twice the principal focal 

6. No real image can be formed by a convex lens when the 
object and the screen are nearer together than four times the 
principal focal length of the lens. 

7. The eye is focused by changing the thickness of the crys- 
talline lens. 

8. A normal eye can not form on the retina clear images of 
objects that are nearer to the eye than 25 cm. This distance is, 
therefore, called the limit of distinct vision. 

9. An eye is far-sighted when the accommodating muscle 
must be used to see distant objects clearly. It is then unable to 


bring the images of near objects to a focus on the retina without 
straining the acconunodating muscle. 

10. An eye is near-sighted when its crystalline lens is too thick 
in the middle. It cannot bring the images of distant objects to a 
focus on the retina. 

11. When an image is formed by a lens, the lens angle of the 
object is the same as that of the image. 

12. The simple microscope enables the eye to focus cleariy 
on the retina the images of objects that are less distant from the 
the limit of distinct vision. The object appears enlarged, because 
it then subtends a larger visual angle. 

13. In the telescope and the opera glass, the light from each 
point of the object is parallel when it enters the objective, and 
also parallel when it leaves the eyepiece. The magnification is 
due to the fact that the real image formed by the objective is 
nearer to the eye combination than it is to the objective, so that 
the lens angle of this image with respect to that combination is 
larger. The telescope and the opera glass therefore give the 
same effect, enlarging the visual angle of the object. 

14. A concave lens causes the light rays to diverge, and forms 
only virtual images. 

15. The practical unit of light is that furnished by a standard 

16. The intensity of the light that falls on 1 cm^ of a surface 
is inversely proportional to the square of the distance of the sur- 
face from the source of light. 


1. If you wish to copy a photograph full size with a lens of 15 cm 
principal focal length, how great must be the distance between the 
picture and the photographic plate? What will be the relation between 
the distances from the lens to the object and image respectively? If 
you wish to enlarge the picture to twice its. size, what will be the rela- 
tion between those distances? If you wish to get the number of cm 
in this case, you must know the relation between the two conjugate 
focal lengths and the principal focal length. If u represent the dis- 
tance of the object from the lens, v that of the image, and / the 


principal focal length, the relation among the three is found to be 

i._j_l. = L. Verify this equation in the case of object and image the 

u V f 

same size (w = v = 2/). ; 

2. Draw a diagram of a convex lens with a principal focal length 
of 4 cm. Draw an object 6 cm from the lens and construct the image 
by the method of Art. 363. What is the relative size of the image? 
Measure the distance from the lens to the image and see if the construc- 
tion verifies approximately the equation in problem 1. 

3. A 16 c. p. incandescent lamp and an arc lamp give the same 
illumination on a screen when the distance from the screen of the in- 
candescent lamp is 10 cm and that of the arc lamp is 100 cm; what is 
the candle-power of the arc lamp? 

4. If the arc lamp in problem 3 takes 9 amperes at 55 volts, 
while the incandescent one takes 0.5 amperes at 110 volts, at what 
rate in watts is energy supplied to each? Which lamp is the more 
efficient? What is the ratio of their efficiencies? 

5. The amount of light that -passes through a camera leps is pro- 
portional to the area of the opening of the lens, i.e., to r^, if r is the radius 
of the opening. If we have two lenses with diflferent sized openings 
of radii, r^ and r2, and if i^ and t2 represent the intensities of illumination 
of the light on a cm2 of the plate at a fixed focal distance /, write the 
proportion which expresses the relation between the intensities and the 
radii of the openings 

^. If we have two camera lenses of the same area of opening but 
of .different focal lengths /i and /2, the same amount of light will pass 
through each under like conditions, but since the focal length of one 
is greater than that of the other, the intensity of illumination of the 
light on one cm^ of the plate will be inversely as. the squares of the 
focal lengths, i.e., ii : ij = /| : f\. Does this explain why a long focus 
lens is "slower" for taking pictures than a shorter focus one of the 
same aperture? 

7. Multiply together the two equations of problems 5 and 6 and 

extract the square root. The result is i : 12 = ^ : ^. What angle is 

measured by ^? Twice this angle, or that subtended by the rim of the 

lens at a point on the plate, is called the angle of aperture. Since 
the "speed" of a photographic lens depends on the intensity of light 
on the plate, may we use this ratio, or its double, the ratio of the diam- 
eter of the lens to the focal length, as a measure of speed? What is 

the meaning of the marks ^, etc., on the stops in some camera lenses? 

402 PHY8ICJ8 


1. If you have a camera, measure the diameter of the stop marked 

and then measure the focal length of the lens — preferably by forming 


an image the same size as the object — ^and dividing the distance be- 
tween object and image by 4. Do you find any relation between the 
things measured and the numbering on the stop? 

2. Take your opera glass lenses out, and measure the focal lengths 
of the objective and eyepiece. This may be done by letting sunlight 
pass through each and measuring the distance at which the sun's image 
is formed by the objective. The eyepiece spreads the light, but its 
focal length may be found by measuring the diameter of the lens, and 
the diameter of the spot of light formed by it on a screen at a measured 
distance. What is the magnification? Determine it by looking with 
one eye at a brick waU and with the other at the same brick wall through 
one tube of the opera glass; you then see two images of the bricks, one 
without the glass the other with it; the magnification is the number of 
bricks in the first image which cover one in the second. You can 
also determine the magnification of telescopes in this same way. Would, 
a plainly marked scale of equal parts be better than the brick wall? 

3. Examine the projecting lantern in the schoolroom and see if 
you understand the operation of its "condenser." Is its projecting 
lens different from a camera lens? Measure the distances from slide 
to lens and lens to screen, and see if the diameter of the slide is to that 
of the image on the screen as the respective distances are to each other. 

4. There are many useful books on photography. You will also 
find Wright, Optical Projection (Longmans, New York), valuable. 
There are some interesting experiments in optics in Hopkins's Experi- 
mental Science. , The best book on light for young students is Professor 
S. P. Thompson's Light, Visible and Invisible (Macmillan, New York). 
This is a series of the Christmas Lectures at the Royal Institution in 
London. It gives a very clear and fascinating account of the latest 
experiments and theories. See also Tyndall's Six Lectures on Light, 
which gives an account of the work of Newton and Young. Tyndall's 
style is a model in clearness and precision, and the charm of the man 
is reflected from every page. 



377. Newton's Experiment. Having studied in the last two 
chapters the way in which light serves us by enabling us to dis- 
tinguish differences in direction, we shall now pass on to the 
discussion of color phenomena. Although color has been ob- 
served, and used extensively from time immemorial, little was 
known concerning the reasons why different substances appear to 
have different colors, until the time of Newton. In 1675, Newton 
discovered that a beam of sunlight when admitted through a small 
hole H insi shutter into a dark room and then sent through an ordi- 
nary glass prism P, was not only bent from its path, but was 
also spread out into a band of various colors, extending from red to 
violet (Fig. 226). He further found that if a second prism, in all 

Fio. 226. Newton's Experiment 

respects like the first, was introduced behind the first in such a 
way that the two together made a thick plate of glass with parallel 
sides, the colored band was reduced to a colorless spot. From 
these experiments Newton concluded that white sunlight is com- 
posed of all the colors of the rainbow, and this conclusion has 
been verified by further investigation. 

Now, this experiment shows that white light is a mixture of 
all different colors in certain proportions, but it does not tell us 



wherein the colors differ from one another. What is the physical 
difference between lights of different colors; for example, between 
red light and blue light? To this question Newton gave no sat- 
isfactory answer, because he did not conceive light to be a wave 
motion. But when we adopt the theory that light consists 
of waves, we are able to form a clear conception of the physical 
nature of differences in color, viz., that the different colored lights 
correspond to waves of different lengths, just as sounds of different 
pitch correspond to waves of different lengths. How can we 
prove that this is so; i.e., how can we measure the lengths of the 
waves of light, in order to find out if they are different for different 

378. Interference Fringes. This may be done in a number of 

ways, but probably the simplest is the following: Carefully clean 

two pieces of the best plate glass, and clamp them together 

so that they touch along one edge and are held apart by 

a fine hair or fiber at the other (Fig. 227). We thus 

have formed between the two plates a wedge of air. If 

now we allow light of one color, like that from a flame 

colored with salt, to fall perpendicularly on these plates, 

we do not see the familiar image of the flame, but, 

instead, there appears a series of bright bands of equal 

width, separated by dark spaces, which are also of equal 

width. These bands are called interference fringes, and 

Am they plainly show that there is something periodic about 

the light, just as the phenomenon of beats makes evident 

the periodic nature of sound. 

No very satisfactory reason can be given for the appearance 
of the black bands unless we conceive that the light consists of 
waves. We have learned in the chapters on wave motion and 
sound that two waves may add themselves together so as to pro- 
duce no motion when their phases are opposite. Similarly, in the 
case of the two plates of glass, we may suppose that waves of 
equal length, but in opposite phases, are thus adding themselyes 
together so as to destroy each other's effects, therefore darkness 



results. Whence do we get the two waves? By reflection from 
the inner surfaces of the two plates. Thus, let PQ and PR, 
Fig. 228, represent these two surfaces^___WSen-the^ 
light falls on the surface PQ, part of iFls^T^flected 
and part passes through. When that which has 
passed through falls on the second surface •Pii, 
part of it is reflected and thus we have two re- 
flected beams, one a d from the first surface, and 
the other b e from the second. Now, it is clear 
that the light reflected at b will, when it reaches c, 
be somewhat different in phase from that reflected 
at a, because the light at c will be some wave 
lengths behind that reflected at a, since it has 
traveled a distance abc more than the light at a. 
Therefore, if this extra distance is half a wave, the 
light at c will be half a wave behind that at a, and 
so the two beams will come together in opposite phases, cancel 
each other's effect, and together produce darkness (c/. Art. 299). 
Similarly, when the extra distance abc is a whole wave, the two 
rays ad and ce will be in the same phase, and so when they are 
added together, they produce light. Therefore, at that place we 
see a bright band crossing the glass. When 
the distance abc is three half waves, dark- 
ness again results, and so on. Therefore the 
successive bright bands occur at places 
where the successive extra distances abc 
traveled by the second beam differ by a 
whole wave, and that the dark bands occur 
when these distances differ by half a wave. 
Thus, in Fig. 229 the bright bands are 
found at the points marked 6, and the dark 
ones at the points marked d. It is clear 
from the figure, that between two bright 
bands the distance from plate A B to 
plate C D increases by one-half of a wave. 
This distance can be measured by deter- 
mining the diameter of the fiber, its distance from the edges of 

Fio. 229. Interference 

OP Waves in the Air 



the plate, and. the distance between the dark and bright bands. 
For example, if the diameter of the fiber is 0.01 mm, and if it is 
60 mm from the place where the plates touch, and if the distance 
between the bands in red light is 2 mm, we may see 30 bands 
on the plate. Since in this case 30 bands correspond to a change 
in distance between the plates of 0.01 mm, one band corresponds 
to a change of ^^^^ = .00033 mm. But this change in distance 
corresponds to half a wave length of red light; therefore 
2 X .00033 = .00066 mm = one wave length of red light. 
There are several other ways of determining the lengths of light 
waves, all of which are vastly more accurate than this. They 
are all based on the principle of interference, and the values 
of the wave lengths obtained by the different methods agree 
closely with one another. 

379. Lengths of the Waves of the Colored Lights. If we now 
illuminate this same pair of plates with green light, the dark bands 
will appear narrower and nearer together — there will be about 40 
on the plate. Therefore the distance between the plates in- 
creases f ^ for each band, i.e., .00025 mm. Therefore one wave 
kngth m green = .00025 X 2 = .00050 mm. Proceeding in a 
similar manner for the other colors, we find the lengths of the 
corresponding waves to be approximately: 

red, .00066 mm 

orange, .00060 mm 

yellow, .00055 mm 

green, .00050 mm 

blue, .00045 mm 

violet, .00040 mm 

We thus find that different colors really do have different wave 
lengths. It is interesting to note how extremely minute the waves 
are. Thus, there are about 2500 blue waves or 2000 green ones, 
or 1500 red ones in one millimeter or 38,000 red waves in an inch. 

380. Interference Fringes in White Light. What will happen 
if we allow white light, instead of light of one color, to fall on the 

COLOR 407 

two glass plates? We find a most beautiful array of many-colored ' 
bands. The colors are not so marked, however, as those in the 
spectrum. These bands are easily produced with an ordinary 
soap solution, such as is used for blowing bubbles. A drinking 
glass is dipped in the soap suds and then set on its side, so that 
the soap film over its end is vertical. As the water drains out 
from between the two sides of the film, a thin wedge of soap solu- 
tion is formed, and the colored bands will be seen stretching hori- 
zontally across the film. The hypothesis that light consists of 
waves enables us to give a simple explanation of the formation 
of these colored bands. For if all the colors are present in white 
light, and if the different colors correspond to waves of different 
lengths, then each color will have a set of bands corresponding 
to it, and these bands will be of different widths. When these 
different sets of colored bands are all present, they overlap, so as 
to give us color mijctures. These mixtures of the different colors 
produce the various tints or tones observed in the two glass plates 
or 'the soap film. 

Thus one of the mysteries of our early childhood is solved; 
for the colors often seen in a crystal or a piece of ice which has a 
crack in it, are formed in this way. So, also, are the colors in a 
soap film, or those seen on the surface of oily water. It is by 
reflection from the two surfaces of the bubble, or the crack, that 
these wonderfully colored interference bands are produced. The 
iridescence of polished shells, and of certain kinds of glass, may 
be accounted for in a somewhat similar manner by the interference 
of waves reflected under suitable conditions from their surfaces. 

381. Dispersion. Having thus learned that different colors cor- 
respond to different wave lengths and that white light is a com- 
position of all the colors, let Us return for a moment to Newton's 
experiment with the prism, and ask how the prism is able to sepa- 
rate the colors and spread them out into a band. Clearly, the 
prism must be acting differently on the different colored 
lights. If we repeat Newton's experiment, and interpose a red 
glass in the beam of sunlight, we find that the path of the red light 
- Js bent a certain amount by the prism, and we get on the screen a 


red spot only. On changing the red glass for a blue one, we 
observe a spot on the screen, but the blue spot does not fall on the 
screen at the same place as was occupied by the red one. On 
examination we find that the prism changes the direction of the 
blue light more than it does that of the red. Thus the prism 
separates the white light into a series of spots of color and the 
entire band of color is thus seen to consist of a series of spots of 
color, each overlapping the adjacent ones. 

The bending of the light is due to the refraction of the glass 
of the prism, and the amount of that refraction is measured by 
the index of refraction, which is the ratio of the velocity of light 
in the glass and in the air (cf. Art. 354). Hence, since the index 
of refraction of blue is found to be different from that of red, 
we conclude that the velocity of blue light in the glass differs 
from that of the red. Since the blue is bent more than the red 
its index is greater, and therefore its velocity in glass must 
be less than that of the red. Thus, we conclude that the separa- 
tion of white light into colors by the glass prism is due to the fact 
that the waves of the different colored lights travel with different 
velocities in glass. This phenomenon of the separation of light 
into colors by a prism" is called dispersion. It is of interest to 
find out whether all transparent substances separate light into 
colors to the same extent that glass does, i.e., how this separation 
depends on the substance of the prism. 

382. The Spectrum. • It would not lead to accurate results if 
we tried to measure dispersion with the apparatus as used by New- 
ton, because each color in the white light forms a spotof color at a 
definite place, and these spots overlap to form the band of colored 
light. Since each spot is larger than the hole (cf. Art. 350), the 
band is fuzzy and not clearly defined. Hence, we must so improve 
the apparatus that we may measure the position of a given spot 
with some degree of accuracy. As you may have surmised, this 
is done by introducing a convex lens L, Fig. 230, in such a way 
that it forms on the screeen images of the hole in the shutter. 
Another improvement is to replace the round hole in the shutter 
by a narrow slit parallel to the edge of the prism, because then the 



successive colored images of the hole do not overlap as much as 
do the round images, as shown in the figure. When we make 
these changes we find that the appearance of the band of color, 

Fig. 230. The Spectrum is Clear and Brilliant 

the SPECTRUM, as it is called, is much improved. The edges are 
now clearly defined and the colors in the center are both brighter 
and more distinct. 

383. Bright Line Spectra. However, one other question needs 
settlement before we can compare dispersions, viz.: What colors 
shall we compare; for the spectrum appears to contaii^ all sorts 
of reds, greens, etc.? We must, then, select some particular ones 
for comparison. W^hich shall they be? If, with the new arrange- 
ment of the apparatus, we interpose a red glass, we find that even 
now the spectrum of the transmitted light is not sufficiently well 
defined to admit of definite measurement; for the red spot, though 
rectangular, is broad and contains many different shades of red, 
i.e., is not monochromatic. Nature has furnished us with a means 
of producing suitable monochromatic colors, for we find that if 
we burn a metal, sodium, for example, the light is nearly mono- 
chromatic. Thus, if we send the sodium light through our appa- 
ratus — a SPECTROSCOPE, let us call it — we note that the spectrum 
consists mainly of a clearly defined bright yellow image of the 
slit through which the light passed before* entering the prism. 
Since this image is a clearly defined yellow line, w^e may measure 
its position with great accuracy. In a similar way, if we burn 
mercury, we find that the corresponding spectrum consists of a 


bright yellow line, a bright green line, and a violet line. Zipc, 
in the same way, gives a spectrum consisting of a red, a green, 
and two blue lines. Other metals give still other lines. 

Now, the great advantage of these lines is that they are very 
clearly defined, and, so far as we have been able to detect, they 
always have the same colors; i.e., the wave length corresponding 
to a given line is always the same in air. Therefore they furnish 
convenient points of reference by which to measure dispersion. 

384. Measurement of Dispersion. We are now in a position to 
compare the powers of two different prisms. Let us take, for 
example, one of water and one of glass, both of the same size. 
We first place the water prism in the spectroscope, and using 
mercury light, for example, mark on the screen the positions of 
the yellow, the green, and the violet lines. We then do the same 
with the glass prism, and find that it not only bends each of the 
rays more than the water does, but also that the distance between 
the successive lines is greater. Thus, the glass not only gives the 
rays a greater deviation, but also the dispersion of the rays is 

Another complication now arises, for if we turn the prism 
about a vertical axis, we find that both deviation and dispersion 
vary; i.e., both depend on the angle of incidence of the light on the 
prism. It will be found, however, that there is one position of 
the prism for which the deviation is smaller than for any other. 
Hence we may compare prisms when in this position, which is 
called that of minimum deviation. On setting the prism in this 
position, it will be found that the light enters and leaves it at the 
same angle with the front and rear faces respectively. We may. 
then, compare the powers of prisms when they are in this par- 
ticular position. 

385. Achromatic Lenses. The importance of this discussion 
of deviation and dispersion becomes manifest when we apply the 
principles we have learned to lenses. For it must be clear that if 
the different colors travel with different velocities in glass, then 
when white light passes through a lens, some of the colored beams 

COLOR 411 

will be refracted more than others by the lens, so that the different 
colors will not all come to the same focus. Thus, if the object is 
a star on the axis of a lens, the blue w^aves, being much more 
retarded by the lens than the red, will come to a focus nearer the 
lens than do the red waves. Hence, when we wish to observe the 
image of a star, we find that there is, strictly speaking, no image, 
for at one point there is a red image surrounded by fuzzy blue 
light and at another a blue image surrounded by fuzzy red light, 
and so on for other colors. This phenomenon is called chro- 
matic ABERRATION. It thus becomes clear that unless we can find 
means of correcting this aberration, lenses for fine work are useless. 

Since chromatic aberration is a result of dispersion, it is clear 
that the solution of the difficulty must be sought in a study of that 
phenomenon. We may then ask, Do prisms of the same size, but 
made of different substances, always differ in the deviations which 
they produce, and also in their dispersion? Or may we have two 
prisms that have the same index of refraction for some one color, 
and yet have different dispersions? The answer to this question 
is, of course, only obtained from careful investigation of numerous 
cases. The result of such investigation is that substances have 
been found which produce the same deviation, and yet have dif- 
ferent dispersions. 

In order to understand how this fact may be used to produce 
colorless images, we must first remember that deviation alone 
is what we need for this purpose. Therefore we must find two 
prisms of such nature that they produce deviation without color. 
This is done in a manner similar to Newton's method of recom- 
bining white light by putting two prisms together with their angles 
in opposite directions (c/. Art. 377). But in Newton's experiment 
the two prisms were of the same size and of the same substance. 
Therefore, by introducing the second prism, he canceled not only 
the dispersion of the first, but also its deviation. But suppose 
the second prism is of some other substance, and of such an angle 
that it possesses the same dispersive power as the first but less 
deviating power. Then the dispersions of the two ^vill cancel 
each other and leave a colorless beam, while some of the deviation 
produced by the first prism will remain. 


This method is the one actually used in constructing lenses. 

If you take out a telescope or opera glass objective, you will find 

that it consists of two lenses, one convex, made of crown glass, 

and the other concave, made of flint glass, as shown in 

Fig. 231. These two lenses are. so shaped that their 

powers of dispersion are equal and that they act in 

opposite directions, while the deviating power of the 

convex one is greater. Therefore, in the combination 

the chromatic aberration is corrected, but the lens is 

still able to deviate the rays, £^nd so it forms an image 

that is clear and free from the fuzzy halo of color. 

Fig!^i. We owe the solution of this problem to John Dollond, 

'matIc an English optician, who produced the first achromatic 

^^^* objectives in the year 1757. Newton had been unable 

to find suitable glasses for the constructon of such a lens, and 

had therefore thought that it was impossible. By the discovery 

of new sorts of glass modem investigation has shown that it is 

possible to produce other types of achromatic lenses than the 

one just described. 

386. Spectrum Analysis. Having thus seen how the phenome- 
non of dispersion nearly prevented us from constructing lenses that 
were useful for fine work, let us ask if dispersion is truly useful 
in any way. To this we must answer, yes, indeed; and bring 
forward the following explanation in justification of the answer. 
We have noted, in what has just preceded, that the light from 
burning sodium, mercury, or zinc, when analyzed in the spectro- 
scope, is found to consist of certain well defined colors. Each of 
these colors is pretty nearly pure, i.e., it consists of waves not 
differing much from one another in length. Further, the colors 
emitted by each substance are found to be different, so that each 
element, when burned, produces in the spectroscope a series of 
bright lines that are characteristic of it. Therefore the spectrum 
furnishes us with a means of determining the nature of a sub- 
stance. For if we bum a substance and analyze with a spectro- 
scope the light emitted, we can recognize the lines and so tell 
what the substance burned is. Thus the dispersion of prisms 

COLOR 413 

furnishes us with a powerful and accurate method of chemical 

In this use of the spectrum we are not confined to the study 
of substances of the earth only, for we receive light from the sun, 
the stars, and other heavenly bodies. Because their light can be 
analyzed in this way, we are able to discover the composition of 
these bodies. So the dispersion of prisms enables us to extend 
our chemical analysis to the farthest visible regions of the uni- 
verse. The results of such study are most interesting, for we find 
that many heavenly bodies produce spectra consisting of bright 
lines, while others produce spectra that are of the same nature 
as that of our sun. So we are able to find out that some of the 
heavenly bodies are composed of burning metals, while others 
are more like our sun. 

387. Continuous Spectra. What is the difference between 
a spectrum consisting of bright lines, and one similar to that of 
the sun, which seems to contain all possible colors? We can find 
out by experiment; for if we heat a metal, say a piece of zinc, in 
a flame, it grows gradually hotter. As we have learned in Art. 
153, it sends out long heat waves at all temperatures, and as it 
grows hotter, shorter waves are added to the complex mass of 
waves, until, when the temperature reaches about 520° C, it begins 
to send out red light waves. If its light is then passed through 
the spectroscope, the spectrum will be found to contain mainly 
red waves. But as the temperature increases, the shorter waves 
appear, until the entire spectrum is produced on the screen. The 
zinc is then said to be white hot. But it has not yet melted. On 
examining the spectrum thus formed, it will be found to be con- 
tinuous, i.e., to contain all the wave lengths that correspond to 
the different visible colors. 

But if the zinc is heated still further until it catches fire and 
bums, i.e., forms a luminous vapor, the spectrum will suddenly 
change to one consisting of the bright lines characteristic of the 
zinc. Thus it appears that incandescent solids produce con- 
tinuous spectra, while incandescent vapors are the source of the 
bright line spectra. Applying this fact to the spectra of the 


heaveniy bodies, we see that if those spectra consist of bright lines, 
we are justified in concluding that the heavenly bodies that pro- 
duced them consist of incandescent vapors; while if the spectrum 
appears continuous, the body must be more like an incandescent 

388. Dark Lines in the Spectrnm. Is our sun a vapor or a 
solid? On examining the solar spectrum carefully, it will be 
found to be neither continuous nor yet to resemble a bright line 
spectrum; for though it appears continuous when the dispersion 
used is small, a higher dispersion reveals the fact that it is crossed 
by a large number of black lines. What do these black lines mean? 
Evidently, that certain wave lengths are not present in the solar 
light when it reaches us. A comparison of the position of these 
dark lines wdth those emitted by burning metals shows that the 
position of the two sets in the spectrum are the same. Thus, for 
example, there are found in the solar spectrum black lines at the 
same positions that are occupied by the bright lines of burning 
sodium. Similarly, for the other metals. What, then, can be the 
nature of the sun that it sends us all other vibrations excepting 
those that the substances we know on earth emit? Is the sun made 
of substances that are totally different from those of the earth? 

The explanation of the presence of the dark lines in the solar 
spectrum was first given by Kirchhoff in 1859, and rests on the 
principle of resonance explained in Art. 314. We there learned 
that bodies that emit powerfully waves of a certain period are 
set into violent vibration when waves of the same period are im- 
pressed on them. We have found a marked example of this in 
sound (c/. Art. 342), for there we discovered that a body may be 
set into violent vibration by resonance, when its natural period 
of vibration agrees with the impressed period. Further, when a 
body is thus set into vibration by resonance, it absorbs the radiant 
energy that falls on it, begins vibrating, and thus becomes itself 
a new source of waves w^hich spread out in all directions about 
it. Thus, the energy of the waves which are traveling in the direc- 
tion of the resounding body is scattered by that body in all direc- 
tions, and therefore less of it is passed on in the original direction. 

COLOR 415 

Now, KirchhoflF reasoned in a similar manner concerning iight 
waves. For he said, ''Since the sodium particles have natural 
periods of vibration which correspond to the bright lines pro- 
duced by their radiations in the spectrum, they must be able to 
vibrate by resonance whenever they are acted on by waves whose 
period agrees with their own. Further, when they are thus set 
into vibration by resonance they must absorb the energy of the 
incident waves, and scatter it in wave motion in all directions. 
Therefore the black lines in the solar spectrum, which indicate 
the absence of the waves that agree in period with those emitted 
by sodium, show that there are, between us and the source of the 
solar light, particles of sodium which absorb the sodium vibra- 
tions from the sunUght, and scatter them in all directions.*' 

389. The Suii'9 Atmosphere. Where can such particles be? 
They do not exist in the earth's atmosphere. They do not exist 
free in the cold regions of space, since they must be in the form 
of a hot vapor in order to execute their free vibrations. They 
must, therefore, be in the solar atmosphere. So we reach the 
conclusion that sodium vapor is a constituent of the sun's atmos- 
phere. Reasoning in like manner, we conclude that the other 
elements, whose bright lines coincide with dark lines in the solar 
spectrum, must exist as vapors in the solar atmosphere. Careful 
study of the bright lines of elements and of the dark lines of the 
solar spectrum shows that almost every known bright line has a 
corresponding dark line there; and, therefore, we conclude that 
the atmosphere of the sun is composed, not of substances different 
from those in the earth, but of the same materials. The lines 
in the spectra of the stars also agree in position with those of 
substances known here. So we conclude that all bodies in the 
visible universe are composed largely of the same substances as 
are found on the earth. 

390. Complex Colors. In Art. 379 we found that colors 
correspond to the lengths of the light waves, and that simple 
colors are those that are produced by waves of a definite length. 
We have learned that incandescent solids send out white light 


consisting of all imaginable visible colors, while an incandescent 
vapor sends out certain simple colors- characteristic of it. We 
have seen how particles of definite natural periods of vibration 
absorb and scatter the energy of waves whose periods of vibration 
agree with their own. We are now ready to enter* on the investi- 
gation of the reasons for the appearance of color in the ordinary 
objects about us. Nothing is more familiar to us than that differ- 
ent objects, illuminated by the same sunlight, appear to have 
different colors. Why do the leaves appear green, the violets 
blue, the goldenrod yellow, etc.? 

The complete answer to this question is at present unknown. 
However, in the light of the principles just learned, and with 
the aid of the spectroscope, we can give a partial answer. If we 
pass sunlight through the spectroscope, and then interpose various 
pieces of colored glass in the path of the light, we note that the 
resulting spectra are all different. Thus, when red glass is 
introduced, the yellow, green, and blue of the spectrum are ab- 
sorbed, while the red and some orange come through. When 
green glass is introduced, the red and the blue disappear, while 
the green and some yellow come through. It will, however, be 
noted that this soii of absorption differs from that which produces 
the dark lines of the solar spectrum. For in the case of absorp- 
tion by metallic vapors, very definite colors are absorbed — what 
we have termed simple colors, which correspond to one particular 
wave length. But the colored glass absorbs a vast number of 
wave lengths and lets another vast number pass through. So 
we see that though the color of the glass, red, for instance, is due 
to the absorption of part of the waves in the sunlight, yet a large 
number of different waves are transmitted, so that the light 
which is thus sifted by the glass is still very far from simple. 

391. Colors of Ordinary Objects. We thus see that the 
colors of ordinary objects are complex in the same way that sounds 
are, in that both consist of a large number of different waves. 
There is, however, this distinction, that the various notes in a 
complex musical sound are related by simple numerical relations, 
while among the waves that produce ordinary colors, no such 

COLOR 417 

relations exist. Consequently we have no color scale, i.e., we 
have no standard universally accepted as to harmony and discord 
of colors. Many attempts have been made by metaphysicians 
to construct a color scale which should correspond to the musical 
scale; and Newton's statement that the spectrum consists of the 
seven colors, violet, indigo, blue, green, yellow, orange, red, has 
frequently been used for this purpose. We are now able to see 
that such attempts are entirely artificial, since there is no simple 
numerical relation between the component parts of a complex 
color, and so our judgment of harmony and discord of colors has 
no such physical basis as has our recognition of tonal relationship 
in music. 

392. The Eye. We can, perhaps, make this clearer if we 
compare the mechanism by which the eye detects colors with that, 
by which the ear detects sounds. Experiments in the reproduc- 
tion of colored pictures have shown that most of the colors recog- 
nizable by the eye can be produced by combinations of red, green, 
and blue light. This fact has led to the theofy of Young and 
Helmholtz, that there are on the retina of the eye three sets of 
nerves, one sensitive to red, one to green, and one to blue. These 
three nerves must be of such a nature that exact coincidence 
between their natural period and the impressed period is not 
necessary in order to make them respond to the action of the waves. 
Our perception of the colors, then, probably depends on the rela- 
tive intensity of the excitation of these three sets of nerves. The 
ear has a large number of nerve fibers, each tuned to a different 
note, therefore it resolves complex tones into their components 
and hears the components separately. The eye, on the other 
hand, has three different sets of fibers, of which a large number 
are tuned to red, another large number to green, and still another 
to violet blue. Therefore red, green and violet lights, acting 
together stimulate all three sets of nerves and give the sensation 
of white. Blue and yellow together do the same, while green 
and violet lights together give peacock blue, a tint between green 
and violet, red and violet give purple, red and green give yellow, 
.and so on. 


393. Paints and Dyes. The action of paints and dyes is 
similar to that of colored glass in one way, but different in another. 
It is different in that the light we receive from painted or dyed 
surfaces is reflected, not transmitted. It is similar, in that the 
light that comes to us after reflection is w^hat remains of the inci- 
dent light after some of its colors have been absorbed. Thus, red 
glass absorbs most of the blue, green, and yellow from white light, 
and transmits only the red and some of the orange. In like man- 
ner, red paint absorbs most of the other colored lights, and returns 
only the red and orange by reflection. 

That this is the action of paints and dyes is shown by the fact 
that the colors of substances appear so different when viewed by 
gas light and in the daylight. For daylight contains all the differ- 
ent colors in large amounts, while gas light lacks much of the 
blue and violet. Since the pigments can send back only certain 
waves from among those which they receive, the blues and violets 
appear dvM in gas light; and in a red light they appear nearly or 
entirely black, because they absorb all the reds, and there are in 
the red light no waves that they are able to reflect. 

394. Mixing Lights and Mixing Pigments. Why is it that 
if we mix two colored lights, say blue and yellow, we see a white 
mixture on a white screen; but if we mix blue and yellow paints 
or dyes, and color the screen with the mixture, the result is green? 
The answer is easily obtained by considering carefully the action 
of the painted and the unpainted screen on the light. The screen 
is white in daylight, because it receives and reflects a mixture of 
all colored lights. If we send two beams of colored light, one 
yellow and the other blue, to the same point on the screen, both 
beams are reflected and make their impressions at the same place 
on the retina of the eye. The result is that all three sets of nerves 
at that place in the eye, the red, the green, the blue, are sufli- 
ciently excited to give us the impression of w^hite. 

The sensation arising from the light reflected by a mixture 
of blue and yellow paint is green, because the blue pigment ab- 
sorbs all the incident light except the green and the blues, which 
it reflects; while the yellow paint absorbs all but the orange. 

COLOR 419 

yellow, and green, which it sends back. Hence, when the two 
pigments are mixed, the mixture returns only those colors that 
are not absorbed by either the yellow or the blue, i.e., it reflects 
the greens only. 

395. Complementary Colors. If we look intently at a brightly 
colored blue spot for several minutes, and then turn our gaze to 
a white wall, there appears to be on the wall a yellow spot of the 
same sizie and shape as the blue one. In like manner, if the 
spot is pink, that which appears to be on the wall will be pale 
green. But we have just seen that a mixture of yellow and blue 
light produces white. Hence these colors are said to be com- 
plementary. This phenomenon of complementary colors may 
be accounted for with the help of the Young-Helmholtz theory of 
color vision. For when we look at a brightly colored spot, the 
nerves sensitive to that color become tired, so that when we look 
at a white surface, which is sending out all colors, the ''blue" 
nerves do not respond, and the sensation corresponds to white 
with the blue left out, i.e., it is that of the complementary color, 


1. White light is a mixture of a vast number of waves of 
different lengths. 

2. Difference in color corresponds to difference in wave length, 
the red waves being longer than the blue. 

3. The wave lengths may be measured by means of the inter- 
ference fringes. 

4. The velocities of different colored lights in transparent 
substances, like water and glass, are different. 

5. The separation of composite light into its colors is called 

6. Dispersion makes the formation of clear images by a 
single lens impossible on account of chromatic aberration. 

7. Chromatic aberration may be corrected by properly com- 
bining two lenses of equal and opposite dispersive powers, but 
of unequal deviating powers. 


8. Every incandescent vapor sends out waves of definite 
lengths which correspond to a few simple colors that are charac- 
teristic of it. 

9. Substances may be analyzed by the spectroscope. This 
analysis may be extended to include the sun and stars. 

10. Dark lines in the spectrum are due to the absorption of 
definite waves by metallic vapors. 

11. The atmospheres of the sun and other celestial bodies consist 
of the simple substances known on the earth. 

12. The colors of ordinary objects are very complex. They 
absorb large numbers of waves of certain .lengths and reflect 
waves of certain other lengths, which give them their character- 
istic tints. 

13. There is no simple numerical relation between the vibra- 
tion numbers of the components of a complex color, and so we 
have no color scale and no physical basis for judging concerning 
harmony and discord of color. 

14. The eye seems to have three sets of nerve fibers, sensitive 
respectively to red, green, and violet blue. 

15. The ear analyzes a complex tone and hears the elements 
separately. The eye receives one sensation as the resultant of 
the action of a complex color. 

16. Two colors that produce the sensation of white when they 
are mixed together are said to be complementary. 

17. The results of mixing colored lights are quite different 
from those of mixing colored pigments. 


1. How may we prove that white light consists of a Idrge number of 

2. How may we prove that different colors correspond to waves of 
different lengths? 

3. What is meant by dispersion? How does a prism deviate a 
beam of light? How does it separate it into colors? 

4. What is chromatic aberration? How is it corrected in lenses? 

5. How is dispersion used for chemical analysis? What improve- 
ments do we have to make in Newton's apparatus in order to produce 
a clear spectrum? 

6. Are the colors emitted by incandescent vapors simple? 

COLOR 421 

7. What is the difference between the spectrum of an incandescent 
solid and that of a vapor? How does this difference depend on the 

8. How does resonance enable us to explain the dark lines in the 
solar spectrum? 

9. How is the sun's atmosphere analyzed? 

10. How is the complexity of ordinary colors different from that of 
a musical tone? 

11. Why do we have no color scale to correspond to the musical 

12. How does the eye detect differences in colors? Has it, like 
the ear, a separate nerve for each separate number of vibrations? 


1. When salt is burned in alcohol, the light emitted by the flame is 
the characteristic yellow light of sodium. Why do people appear 
ghastly in this light? If a yellow gas flame, which has a similar ef- 
fect, is surrounded with a red glass globe, does it make .the people 
appear more natural? Why? 

2. If a piece of blue paper is illuminated with red light, what color 
does the paper appear to have? What color does it appear to have 
when illmninated with yellow light obtained by passing daylight 
through yellow glass or a solution of bichromate of potash? If the 
yellow light were that of sodium, what would be the result? 

3. When you look through a prism at a window or a broad band 
of white paper, you see a broad patch of white with red, orange and 
yellow on one edge, green, blue and violet on the other; but if you 
look at a narrow slit through which white light is coming or at a 
narrow band of white paper, you see a continuous spectrum. Explain 
the cause of the difference. 

4. Why is it that the complementary of any one of the spectrum 
colors is a complex tint and not a pure color? 

5. Why do interference fringes in white light give complex tints 
instead of pure spectrum colors? 


1. Make a soap solution such as you use for blowing soap bub* 
bles, and form a soap suds film over the open end of an ordinary 
drinking glass with straight sides, not tapering. Set the glass on its 
side so that the film stands vertically. See if interference bands ap- 
pear as in the air wedge mentioned in Art. 378. If the bands do not 
appear, the solution is too strong, if the film breaks, the solution is 
too weak. Does a black spot appear at the top of the film? Can you 


find out how thick the film is at the bottom? Remember that the 
thickness of the fihn increases half a wave length for every black band. 
If you view the film in red light, and the wave length of the red 
light in air is 0.000065 cm, how long is it in water, i.e., in the soap 
solution if the index of refraction is 1.33? 

2. Can you find out how the rainbow is formed? How does the 
light pass into and out of each drop of water so as to be separated 
into its colors? Why is the bow always curved? Has the' rainbow 
an end? Why do we sometimes see two bows? 

3. Consult a book on botany and see what you can find out about 
chlorophyll and its relation to the green color of leaves. What rays 
does it absorb, and which does it reflect? 

4. You can buy a small prism from an optician for about 30 cents, 
and with it make a number of interesting and instructive experi- 
ments. See Twiss' Laboratory Exercises in Physics (Macmillan, N. Y.) 
Exercise 43. This will suggest others; e.g., examine with the prism, 
as there directed, narrow strips of the Milton Bradley colored papers, 
which may be obtained at your bookstore, and the lights from red and 
green fire, which you can buy at your druggist's. 

5. The Milton Bradley color top, which can be bought postage 
prepaid for 6 cents, is an endless source of amusement and instruction. 
By all means get one and use it as directed in the Bradley color book, 
Hopkins' Experimental Science, or Mayer and Barnard's Light. 

You will find the following books of interest in connection with 
the subjects discussed in this chapter: O. N. Rood, Modem Chromat- 
ics (Appleton, N. Y.); N. Lockyer, Spectrum Analysis (Appleton, N.Y.) ; 
Vanderpoel, Color Problems (Longmans, N. Y.); Elementary Color {MilUyn 
Bradley Co., Springfield, Mass.); Mayer and Barnard, Light (Appleton, 
N. Y.)- This is full of beautiful home experiments. Thompson's 
Light, Visible and Invisible, Lommel's Nature of Light (Appleton, N. 
Y.), and Professor D. P. Todd's New Astronomy (Am. Book Co., Cin- 
cinnati) , also contain much that is interesting and not difi^icult to read. 


396. The Medium. In the preceding chapters we have 
assumed that light is a wave motion. We have learned how, by 
ordinarily traveling in straight lines, it enables us to judge the 
directions of the objects. We have then discovered some of the 
consequences of considering that differences in color correspond 
to differences in wave length. We now pass to the problem 
of determining more exactly the nature of the light waves. If 
light is a wave motion, in what medium does it travel, for we 
have learned that a medium is necessary for the propagation of 
w^avfes? It is not air, because light comes from the filament of 
an incandescent lamp even though the air is entirely pumped 
out of the bulb, as was noted in Chapter VIII. Also, light comes 
from the sun and stars to us, although we are certain that our 
atmosphere does not extend that far. But how shall we find out 
more about the medium in which light waves are propagated? 
How determine its properties? One possible way is by determin- 
ing the velocity of light, as we determined the properties of air 
from a study of the velocity of sound in it (c/. Art. 313). But how 
shall we measure the velocity of light? It is well known that its 
velocity is very great; for, as was noted when discussing the velocity 
of sound, light appears to travel short distances instantaneously. 

397. Galileo's Method of Measuring the Velocity of Light. 
The first to propose a method of measuring the velocity of 
light was Galileo. He suggested stationing two observers on 
distant hills and supplying each with a lantern, fitted with a shutter 
that could be closed and opened quickly. Then observer 1 opens 
his lantern; when observer 2 perceives the light, he opens his. 
Then observer 1 has but to note the time thr^t elapses between 
opening his lantern and seeing the flash of t^ e other lantern, 

. 423 



and this time should be that taken by the light to pass from 
observer 1 to observer 2 and back. So reasoned Galileo. The 
experiment was tried, but without satisfactory results, for it was 
found that the time was too short to measure accurately. Further, 
the method is inaccurate, because there is introduced the time 
taken by observer 2 in becoming conscious of the appearance of 
the light from observer 1 and opening his lantern. This time is 
greater than that taken by the light in traveling the entire dis- 
tance. The experiment is correct in principle, but the application 
of the principle can be improved. 

398. Fizeau's Method. The first improvement consists in 
replacing .observer 2 by a mirror. Since the action of the mirror 
is automatic, this change eliminates the inaccuracy due to the 
operations of observer 2. But even then it is found that observer 
1 sees the light before he can get his lantern fully open, i.e., he 

is too slow for the light. 
Therefore we have to 
replace the shutter on 
his lantern by a 
mechanical device that 
will open and shut 
the lantern more 
quickly. This was 
done by Fizeau in 
1847. The principle 
of his experiment is illustrated in Fig 232. Light from the source 
S passes through the hole h in the box and thence to the distant 
mirror M. It then retraces its path and is partly reflected at 
the glass plate G to an observer at E. On one end of the box is 
a toothed wheel W, which is so placed that when it revolves the 
light is cut off whenever a tooth t of the wheel is in the path of 
the light, and allowed to pass whenever a notch n is in that 
position. As the wheel rotates, the light is alternately cut off 
and let through. By rotating the wheel rapidly we are able to 
make these openings and closings of the lantern follow one 
another as rapidly as desired. 

Fig. 232. Pkinciple of Fizeau's Experiment 


But it is still necessary to be able to measure the time accu- 
rately. This may be done by the notched wheel itself, for if 
we arrange matters so that the observer sees through the notches 
of the wheel the light reflected from the distant mirror, the re- 
turning light will be cut off if the time taken by it in traveling 
from the wheel to the distant mirror and back is the same as that 
taken by the wheel in turning, so that a tooth takes the place pre- 
viously occupied by a notch. Now, it is easy to measure the 
angular velocity of the wheel, and also to count the number of 
notches in its rim; we may, therefore, determine accurately the 
time taken for a tooth to replace a notch. Twice the distance of 
the mirror divided by this time will then be the velocity of light. 

The experiment has been made many times in this and in 
other ways, and the result is that light has been found to travel at 
the rate of 186,000 ^^ = 3 x 10'* ^. Since the circumference 
of the earth is about 25,000 miles, it appears that light is able 
to traverse in one second a distance equal to about 7 times the 
circumference of the earth. 

Though this seems a surprisingly great velocity, it is not so 
tremendous when we consider the distances from the earth to 
the sun and the stars. Thus, it has been found that it takes 
about 8 minutes for light, traveling at this rate, to come from the 
sun to the earth; and the distance of the nearest star is so great 
that it takes light over three years to travel from it to the earth. 
Further, the most distant stars that we can see are so far away 
that it takes light some 5,000 years to come to us from them, and 
probably there are stars still farther away. These figures show 
us that the ratio of the circumference of the earth to the distance 
of a faint star is that of \ sec to 5,000 years, or TTiJTTrTi^iiirTri^TrTr- 
Thus the knowledge of the velocity of light helps us to form some 
idea of the immensity of space and of the relative size and impor- 
tance of the earth in comparison with the universe about it. 

399. The Ether and Its Properties. But what of a medium 
that can propagate waves at the rate of 3 X 10'° ^? Can we 
apply equation (15) Art. 296, which gives the relation between 
velocity, elasticity, and density? If we can, it is evident that the 


elasticity must be enormously great and the density exceedingly 
small in order that the square root of the quotient may be so 
large a number. But how can we measure the elasticity of this 
medium? In the case of air the elasticity is easily measured 
by compressing the air and measuring its change in volume. 
Can we apply pressure to the medium in which light travels? 
Can we measure its density? These quantities are evidently so 
small that we can not measure them by mechanical means; for 
when we have pumped, the air out of an incandescent lamp bulb 
as far as is possible, our gauges indicate no measurable pressure, 
and we have not been able to weigh or measure the medium in 
* finy mechanical way. But though we are not able to determine 
the numerical value of these factors for the medium, we can give 
it a name. So we call it the ether. We shall have occasion to 
learn more of the properties of the ether as we proceed. At 
present we can only conclude that whatever properties it may 
have, it does not react in any measurable way to the ordinary 
mechanical forces, such as pressure, torsion, and the like. 

Since we thus learn that we can not measure the properties 
of ether as we can those of air by mechanical means, we are com- 
pelled to seek elsewhere for some method of finding out what its 
nature is. Are there not other phenomena which involve the ether, 
and by means of which we may study its character? We have 
already noted, in the study of electricity, that electricity and mag- 
netism do not act by means of air, but through some other me- 
dium (Art. 286). This fact has led scientists to investigate 
carefully whether those properties of the ether which may be 
discovered by means of experiments in electricity and magnetism 
can in any way assist us in framing a theory concerning the prop- 
erties of light waves. 

400. Electric Waves. When a Leyden jar is discharged, 
the electric discharge vibrates very rapidly back and forth a 
number of times between the terminals (cf. Art. 197). There- 
fore we have in the electric spark a vibrating something which 
may send out waves. Can we detect at a distant point the 
discharge of a Leyden jar in any other way than by the sound 



Fig. 233. Electrical Resonance 

and light of the spark? If we place a second jar, in every way 
like the first, in close proximity to the first, we notice that a spark 

passes between its 

terminals at S, Fig. 
233, whenever one 
passes between the 
terminals of the first 
jar. We must then 
conclude that the 
second jar has electric 
oscillations generated 
in it by resonance. 
But resonance implies 
waves; and so we con- 
clude that the spark, probably, is a source of electric waves. 
These waves, as is now well known, can also be detected in other 
ways; in fact, they are the waves with which wireless telegraphy 
operates, for wireless messages are sent by discharging electric 
sparks. So we learn that electric sparks generate electric waves 
which travel to distant points and may there be detected. 

If electricity acts by means of the ether, these electric waves 
must be ether waves. Therefore it is of interest to find out how 
fast they travel, because this knowledge will enable us to com- 
pare them with light waves. We can measure their velocity by 
means of the equation v = nl (Art. 296), since the number of 
vibrations can be calculated from the dimensions of the jar used 
in sending the sparks; and the wave length can be measured by 
causing the impulses to form stationary waves on long wires, and 
measuring the distance between the nodes (Art. 301). The 
first experiments of this nature were performed by Hertz in 1888, 
and his result ushered in a new epoch in physical science, for he 
found that these electric waves travel with the same velocity as 
light does. 

The conclusion which we draw from this remarkable result 
is that the ethers of light and of electricity are, the same; and, 
further, that light is an electric vibration, not an elastic one. 
Since the announcement of this admirable theory, many other 



facts have been discovered that all add weight to the argument, 
and make us more content with the simplicity and the elegance 
of the conclusion. 

401. WireleiB Telegraphy. When these electric waves are 
used in transmitting messages without wires, they are started by 
the spark from an ordinary induction coil. The receiving appa- 
ratus (Fig. 234) is very interesting. It consists of a small glass 
tube C, to which are fitted two metal plugs j> p. The ends of 
these plugs are about 1 mm apart, and the space between them 

Fig. 234. Diagram of receiving station for wireless messages 

is filled with loose nickel or silver filings. A current from a small 
battery b passes through this tube^and the relay R, and back to the 
battery. When the circuit is thus completed, very little current 
flows, because the resistance of the loose filings is large, and 
therefore the relay armature is not moved. But when electric 
waves fall on the filings, their resistance is in some way diminished. 
The current through the filings and the relay is thus increased, 
and the relay armature is pulled toward the magnet. This action 
of the filings is called coherence, and the tube C is called a co- 
herer. When coherence takes place, the relay closes a local 
circuit in which a sounder S is operated by a battery B, as in 
the ordinary telegraph system. 

In order to break the circuit through the coherer, it must be 
tapped so that the filings are shaken up. It is, therefore, usually 
placed so that the armature of the sounder strikes the coherer 


when it flies back. The sensitiveness of the apparatus may be 
much increased by connecting one end of the coherer to earth and 
the other to a long wire stretched vertically on a high pole. Such 
a vertical wire is called an antenna. In Uke manner, the power of 
the sending coil may be increased by attaching one of its termi- 
nals to an antenna and the other to earth. A Leyden jar across 
the tei-minals of the coil often helps. The points at which the 
coherer and the sounder circuits are broken must be shunted 
with coils of fine wire of high resistance, for if a spark occurs when 
a circuit is broken, it sends out waves that affect the sounder and 
confuse the signals. 

402. The Complete Spectrum. We are now in a position to 
enlarge our ideas of the spectrum. Since the electric waves travel 
with the velocity of light, and since their numbers of vibrations 
vary from about 1000 to 6 X 10^^ per second (c/. Art. 197), their 
wave lengths vary from 3 X 10^ to 0.5 cm. In Art. 397 we learned 
that a long red wave has a wave length of 0.000066 cm and a short 
blue one a length of 0.00004 cm; we now find that the electrical 
vibrations send out waves of greater length than those correspond- 
ing to red light. We have also noticed that as a body is heated, 
it sends out first longer heat waves, then shorter heat waves, and 
finally, at a temperature of about 520*^ C, it begins to emit red 
light waves. Putting these three facts together, we may conclude 
that the entire spectrum is much longer than appears to the eye, 
for it must contain heat and electrical waves beyond the red. 
This portion of the spectrum is therefore called the ultra-red, 
or heat spectrum. Investigation shows that this spectrum really 
exists; for if we place in the solar spectrum beyond the red an 
instrument capable of detecting heat — for example, a thermometer 
— we shall find that it indicates heat action there. 

Further, if we take a photograph of the visible spectrum, 
we find that the photograph indicates the presence of waves 
beyond the violet end; i.e., we can photograph more than we can 
see. So we conclude that there are electric waves shorter than 
those corresponding to violet light, and that these waves can act 
on a photographic plate. This extension of the spectrum is called 


the ULTRA-VIOLET, OF photo-chemical spectrum. The shortest 
of these photo-chemical waves that has been measured, has a wave 
length of .00001 cm. 

We thus see that the entire spectrum contains a large number 
of waves in addition to those that produce the sensation of 
sight, for it contains waves varying in length from 3 X 10^ 
cm to those of length .00001 cm. Of these waves, those lying 
between the limits 3 X 10^ and .5 cm are called electric waves, 
and are able to act on electrical apparatus; those between the 
limits .008 and .00008 manifest heat action, while those between 
the limits .00008 and .00004 affect the eye and are called light. 
Those that are shorter than .00004 cm may be detected by their 
action on a photographic plate. It is possible that waves exist 
beyond these limits, but they have not been detected by any 
of the devices at present known to science. It is the present belief 
of scientists that all these waves are of the same nature, i.e., that 
they are all electric waves. 

403. What Spectra Tell Us. The detailed study of spectra 
is of great importance in physics, not only because we can thereby 
analyse chemical compounds, but also because it opens a pos- 
sible way of discovering the nature of the vibrating particles 
which are the sources of light waves. The principles on which 
this study is based have been discussed in what has preceded, 
but it may be well to bring them together here for the sake of 
showing their relations. Thus, we have seen (Art. 340) how a 
vibrating string sends, out waves whose lengths are related by the 
simple ratios, 1, i, J, etc. Conversely, when we receive such a com- 
plex sound wave, and on analysis fin(J that its component vibra- 
tions are related by these simple ratios, we are justified in con- 
cluding that the source of the wave is either a vibrating string, a 
rod, or some other body which is capable of sending out that 
particular series of overtones. Now, all bodies that send out 
such a series of overtones are long and thin, i.e., they resemble a 
straight line in geometrical form. So when we have analyzed a 
complex wave and found it to consist of a series of simple waves, 
whose lengths are related by the ratios, 1, J, J, etc., we conclude 


that the geometrical form of the vibrating source of the waves is 
a straight line. 

If, however, the complex wave is found, on analysis, to con- 
tain the overtones characteristic of a bell, we are justified in con- 
cluding that the vibrating body is shaped like a bell; and, simi- 
larly, if the component vibrations are those characteristic of a 
solid ring, or of a sphere, or of a body of any other particular 
geometrical form, we infer that the source of the waves has that 
particular form. A similar conclusion may be drawn concerning 
the geometrical form of a vibrating source of light; for we have 
learned how each incandescent vapor sends out a certain series of 
simple vibrations characteristic of it. We can sort out these 
simple vibrations by the spectroscope, determine their vibration 
numbers, and so find out whether any numerical relation exists 
between the vibration numbers that correspond to the bright lines 
in the spectrum of any element. Careful experiments have 
proved that such numerical relations do exist among the vibration 
numbers that correspond to the bright lines of many of the chem- 
ical elements. What, then, is the geometrical form of the body 
that can vibrate in such a way as to send out the corresponding 
vibrations? What is the geometrical form of the vibrating body, 
i.e., of the atom? 

To this question science has not yet found an answer; for 
though we know the relations among the vibration numbers of 
the series of waves sent out by the atoms of many substances, 
that relation is very complex, so that we have not yet been able to 
show what geometrical form is capable of vibrating in such a way 
as to send out that series of vibration^. The problem, however, is 
not hopeless; and during the last few years considerable progress 
has been made towards its solution. All that can be said at 
present is, that this study of spectra offers a possible way of en- 
abling us to form some conception of the shape and construction 
of atoms. 

The importance and wonders of spectra thus become clear; 
for we see that from them we can not only discover the make-up 
of stars at almost infinite distances from us, but we can also study 
the mechanism of the tiniest thing that the human mind has been 


able to conceive. We see that we are surrounded by phe- 
nomena of marvelous complexity, yet governed by simple prin- 
ciples — how these phenomena and the principles that govern 
them extend to both the infinitely great and the infinitely small, 
without any conceivable limit. For who can set an outside bound- 
ary to the universe, or who can measure the size of the smallest 
particle which plays its part and has its own use in the 
economy of Nature? Yet throughout this vast complexity of 
relations and interrelations, among infinitely small atoms and 
the infinitely great universe, we can discern the operation of com- 
paratively simple principles, which are manifest in the least, as 
weir as in the greatest, of the operations of the universe. 


1. The velocity of light is 3 X 10^"^. 

2. The elastic constants of the ether can not be measured by 
mechanical means. 

3. The electric spark generates electric waves varying in 
length from 3 X 10^ to 0.5 cm. 

4. These long electric waves travel with the velocity of light. 

5. Heat and light waves are of the same nature as electric 
waves, and the ether is the medium which transmits them all. 

6. The complete spectrum contains electric, heat, light and 
photo-chemical waves. 

7. The spectrum enables us to study the geometrical form of 
atoms as well as the constitution of the heavenly bodies. 


1. How is it possible to measure the velocity of light? 

2. How long does it take light to travel a distance equal to the 
circumference of the earth? How long to come from the sun? From 
a star? 

3. What medium transmits light? Can we measure its elasticity 
by mechanical means? 

4. What other phenomena besides light depend on the ether for 
their manifestations? 

5. How are electric waves generated? How detected? Describe 
a wireless telegraph system and explain its operation. 


6. What is the velocity of propagation of electric waves? How is 
their velocity measured? 

7. Why do we conclude that the waves of heat and light are electric 
in their nature? 

8. What different kinds of waves make up the complete spectrum ? 

9. Can we study the geometry of an atom by the spectrum? How? 

10. What are the limits of the domain within which the study of 
the spectrum is valuable? 


1. A certain Leyden jar is discharged and the oscillations of the 
spark are found to have a period of ToioTV s^c; what is the length of 
the electric wave started by the spark? If the wave length of so- 
dium light is 0.000059 cm, how many oscillations does a sodium par- 
ticle execute per second? 

2. Since incandescent bodies are radiating long heat waves as well 
as light waves, much of the energy supplied to them is wasted, as far 
as light making is concerned. For example, the unit of light is the 
candle, and it has been found that a unit candle radiates light energy 
at the rate of about 19 X 10' ^^. At what rate does a 16 candle in- 
candescent lamp radiate light energy? Such a lamp requires for its 
maintenance to be supplied with energy at the rate of about* 55 
watts; what is its efficiency as a light producer? 

3. In Art. 170 the heat of combustion of illuminating gas was given 
as 18 X 10* gm cal per cubic foot. An ordinary gas flame consumes 
5 j- — and radiates with an intensity of about 16 candles. What is the 
efficiency of the flame sis a light producer? 

4. An arc lamp radiates with an intensity equal to that of 2000 
candles and requires an expenditure of about 500 watts, what is 
its efficiency? 

5. Is it more economical to bum illuminating gas in a gas en- 
gine, and use the engine to run a dynamo, and let the dynamo 
feed an arc lamp, than it is to bum the gas directly for its light? 

6. A cm? of the earth's surface, perpendicular to the path of 

the sun's rays, receives light energy from the sun at the rate of 

7 X 10^ ®— ^. A cm^ distant 1 m from a standard candle, and per- 

sec ^ ^ 

pendicular to the rays, receives light energy from the candle at 

the rate of 15 ^^. How many candles at the distance of 1 m 

would be necessary to give the same intensity of illumination per 

cra2 as is given by the sun? If the distance to the sun is 15 X 
W^ m, how many candles at the distance of the sun would be re- 
quired to illuminate the earth with the same intensity as the sun 



1 . If there is a wireless telegraph station near you, visit it and 
find out how it works. Ask the operator if they are able to tune 
the receiving instrument so that it will respond only to messages 
intended for it. 

2. What can you find out about the efficiency of a fire-fly? 
How do they produce so much light with so little expenditure of 
energy? May the phosphorescence of decaying wood be a similar 

3. You will find interesting reading on the topics of this chap- 
ter in Oliver J. Lodge, Signalling Through Space Without Wires (Van 
Nostrand, N. Y.); R. T. Glazebrook, J. Gierke Maxivell and Modern 
Physics; Lodge, Modern Views of Electricity (Macmillan, N. Y.); and 
Thompson's Light, Visible and Invisible. 

4. It is not a very difficult task, with the help of Profe^ssor Lodge's 
book, to make a coherer and connect it as in Fig. 234, so as to re- 
ceive wireless signals from the sparks of a small induction coil or 
electrostatic machine. Many boys have made their own outfits, in- 
cluding the induction coil, and operated them successfully over con- 
siderable distances. 


404. Origin of Light Waves. In this final chapter we shall 
endeavor to give an outline of the argument on which our present 
theory concerning the nature of the ultimate particles of matter 
is based. What has preceded has made us familiar with the 
main facts on which this argument is founded. 

The first point in the theory is the hypothesis that light con- 
sists of ether waves, and from this we must infer that the source 
of those waves is a vibrating something. Further, since these 
waves are so minute, the vibrating something must be very small, 
so as to be able to vibrate very rapidly — some 6 X 10'* times 
a second. We are thus led to conceive that the sources of light 
waves are minute vibrating particles. 

405. Electrons. Since the waves of light are waves in ether, 
ih^ vibrating something must be of such a nature that it is able, 
by its vibrations, to disturb the ether, and so become a source of 
waves. Therefore, in order to conceive how these minute vibrating 
particles can disturb the ether, we are led to the further hypothesis, 
that each of them carries an electric charge; for we have learned 
that a vibrating electric charge, such as that with which we 
have become familiar in the sparks from a Leyden jar, can start 
electric waves similar to the light waves (Art. 402). The hy- 
pothesis that there are in Nature tiny particles which carry elec- 
tric charges is further justified by the phenomena of electroly- 
sis; for we have learned (Art. 286) that the actions of the ions 
in an electrolyte is described most simply by assuming that they 
carry electric charges. So we have come to believe that light 
waves have their origin in the vibration of minute electrically 
charged particles. These particles have been named electrons. 



Do they manifest themselves in any other way than as sources 
of Hght waves? Are they the same as the ions of electrolysis? 
How large are they? How are they set into vibration? 

406. Crookes' Vacuum Tubes. Taking up these questions in 
order, we may answer to the first that we have good reason to 
believe that the phenomena observed in connection with electric 
discharge in vacuum tubes are due to these electrons. What are 
some of these phenomena? Let us analyze the action in a vacuum 
tube as the air pressure is gradually diminished. Thus, if we 
send the spark from an induction coil through one of these tubes, 
the discharge inside the tube, if none of the air has been pumped 
out of it, will, of course, be exactly like that in air. But if we 
connect the tube with an air pump, and begin to pump the air 
out, the appearance of the discharge changes. It first becomes 
less like the familiar, sharply defined, zigzag flash. Fig. Ill, and 
spreads out in a broad band, giving a diffused purple glow. 
Presently, as more air is pumped out, this light appears to grow 
whiter and fill the tube. As the exhaustion is continued, the 
whiteness disappears, and we notice a pale blue streamer extend- 

Jng from the negative electrode and perpendicular to its surface. 
Since this streamer proceeds from the negative pole or cathode, 
it is called the cathode beam and is said to consist of cathode 


407. Cathode Bays. These rays have been carefully studied 
and found to possess many peculiar properties. We find, in the 
first place, that where they fall on the glass walls of the tube or on 
certain substances placed in their path, they cause these substances 
to emit light. They are then said to produce fluorescence. 
In the next place, the cathode rays travel in straight lines per- 
pendicular to the surface of the cathode, for such fluorescent light 
always appears in the direction perpendicular to the surface of 
the cathode, and the shadow cast by a metal screen is perfectly 
well defined, as shown in Fig. 235. Finally, they are sensitive 
to a magnetic field; for if a magnet is brought near the tube, the 


positions of the fluorescent spots change. In the light of these 

facts, what assumption may we make as to the nature of these 

rays? All the phenomena • 

exhibited by them may be 

simply described if we 

assume that they consist of 

minute particles, carrying 

electric charges, and shot 

out from the cathode by the 

electric force there acting. 

Since the cathode is the 

negative pole, it is clear fiq. 235. Shadow in Cathode Rays 

that the charges carried 

away from it by the particles must be negative. 

It is easy to see how such negatively charged particles, when 
shot from the cathode, would be electrostatically repelled along 
straight lines perpendicular to the surface of the cathode. It is 
also easy to comprehend how they can produce fluorescence when 
they strike, by jarring the smallest particles in the substance 
on which they strike, and so causing them to shiver and send out 
light. But why should the cathode rays be bent from their straight 
path by a magnet? To understand this, we must recall the fact 
that a charged particle in motion produces a magnetic field (Art. 
227). Therefore, if the cathode rays consist of charged particles 
traveling in straight lines, they should produce a magnetic field; 
and if they do so, of course, the cathode beam itself would be 
magnetic and would be repelled or. attracted by an outside 
magnetic field. 

We can thus understand how the phenomena exhibited by the 
cathode rays are simply described by assuming that those rays 
consist of negatively charged particles shot out from the cathode 
and traveling in straight lines. So we have found that in the 
phenomena of the vacuum tube we may be dealing with electrically 
charged particles or electrons. We may then ask whether these 
cathode particles may not be similar to those whose vibrations 
we have conceived to be the source of light waves. We can best 
arrive at a conclusion on this matter if we first find out something 


concerning the size of the electrons in the cathode rays. But 
can we determine their size? 

408. Size of Electrons. Science has been unable to answer 
this question until very recently, for it is no easy matter to de- 
terinine the mass of particles so small that the weight of millions 
of them together could not be detected by our most sensitive balance. 
However, the bending of the cathode rays by a magnetic field 
enables us to approach a solution in the following way: If the 
cathode rays consist of particles each carrying an electric charge 
e and traveling with a velocity v, it is clear that the strength of 
the magnetic field generated by them will be proportional to 
both e and v (c/. Art. 227). Therefore, the force acting between 
the field of each particle and that of the external magnet, //, 
will be proportional to e v II (Art. 205). But this force gives the 
particle a sideways acceleration, and this is clearly proportional 
to the magnetic force e v H, and also inversely proportional to 
the mass m of the particle. Therefore the sideways acceleration 

is proportional to . Now, it is easy to measure //, and it 

has been possible to measure v\ and since we can also measure 

the sidewavs acceleration, we can determine the value of — ; i.e., 


of the ratio of the charge of a particle to its mass. This ratio 
is found to have the numerical value of 1.87 X 10^. The value 
of this ratio is the same no matter of what substance the cathode 
is made. 

Now, we have conceived (Art. 227) that the particles acting in 
electrolysis are charged particles, and that they move under the 
action of the electric forces in the solution. Evidently the velocity 
of their motion will be proportional to the charge e which each 
carries, to the strength E of the electrostatic field, and inversely 
proportional to the mass m of the particle; i.e., their acceleration 

will be proportional to — . It is easy to measure their accelera- 
tion, and also to determine E, and so we reach another value of 
this ratio — ; but this time it is for hydrogen 10*. 


What is the reason for this difference in the two results? Are 
the charges e the same, and the mass of the cathode particles 
less, or are the masses the same, and the charges of the cathode 
particles greater? The answer to this question we owe to the 
skill and ingenuity of J. J. Thomson, of Cambridge, England; 
for he has succeeded in showing by experiments, whose descrip- 
tion would lead us too far from our present argument, that the 
charges in the two cases are the same, and therefore that the 
masses of the cathode particles are much smaller than those of the 
ions with which we deal in electrolysis. But in electrolysis we 
have every reason to believe that the smallest masses of the ele- 
ments involved, i.e., the ions, are the atoms. Therefore we con- 
clude that the cathode particles or electrons are much smaller 
than atoms. 

The relative sizes of the atom and the electron may be ob- 

tained by dividing one value of the ratio — by the other. Thom- 
son thus arrived at the conclusion that electrons are so small thai 
it takes 1870 of them to make a hydrogen atom. Experiments with 

other elements than hydrogen tend to show that the ratio — in 

electrolysis is inversely proportional to the masses of the atoms, 
and therefore we must conclude that the number of electrons 
necessary to produce a mass equal to that of an atom of any ele- 
ment is 1870 times the mass of an atom of that element. This 
conclusion has recently been confirmed by other experiments 
along wholly different lines; so we now believe that an atom is 
composite and contains many electrons in its make-up. 

Thus, at last, science seems to have found in the electron a 
particle which is smaller than the atom and which may be found 
to be the ultimate particle of matter. 

409. Eadio-Activity. But are electrons found free in nature, 
or do we know them only in vacuum tubes? The study of the 
recently discovered phenomena of radio-activity leads us to believe 
that the radium rays consist of the emission, by radium and other 
similar svbstances, of electrons carrying negative charges; for the 


emanations of these substances act in many ways like the cathode 
rays. Probably many of you have seen one of Professor Crookes* 
spinthariscopes, in which a small particle of radium is mounted 
over, a screen of some substance, like zinc sulphate, which becomes 
fluorescent when cathode rays fall on it. On observing the screen 
in a dark room, it is seen to be scintillating all over with tiny sparks. 
These are believed to be produced by the bombardment of elec- 
trically charged particles that are shot out by the radium so as 
to strike the fluorescent screen. This phenomenon and many 
others lead us to believe that electrons exist free in nature, since 
they are emitted by the radio-active substances. 

410. How Electrons Start Ether Waves. We may now re- 
turn to the phenomena of light, and ask how the conception that 
atoms are made up of large numbers of electrons assists us in 
conceiving a mechanism to describe the origin of light waves. 
We have already learned that light waves are brought into ex- 
istence by sufficiently heating any substance. We must then try 
to form a picture of the way in which the heat energy may be con- 
verted into energy of vibration in solids and in gases. The first 
point that must be noticed in this connection is that heat expands 
the body; and therefore, if we conceive the body to consist of 
small particles, these must be separated from one another by the 
action of the heat. Further, we have noticed how all bodies are 
sending out long heat waves at all temperatures (Art. 149), and 
from this fact we must conclude that the small particles of which 
the body consists are vibrating even at the low temperatures. 
Therefore we may imagine that heating a body both increases 
the amplitude of the vibrations of those small particles, and also 
separates them further from one another. 

When these particles form a solid, they are very close together, 
and therefore, when their vibrations are rendered more intense 
by heating, they must collide with one another more frequently 
and with greater energy than at a lower temperature. When 
two particles collide, each receives a shock, which must cause 
its component parts to shiver and send out for a brief instant a 
large number of waves. These waves are not those corresponding 


to the natural periods of vibration of the particles. Such vibra- 
tions are called forced vibrations (Art. 333), and they die out 
very rapidly. But when the particles make up a solid substance, 
the impacts between them take place so frequently that almost all 
the vibrations sent out by a solid are forced vibrations. There- 
fore the spectrum of a solid sending out vibrations under these 
conditions, contains all possible wave lengths, and so it is a con- 
tinuous spectrum. We may thus draw a mental picture of the 
mechanism by which continuous spectra are formed. 

But when the particles are separated by considerable dis- 
tances, as they are when the substance becomes a vapor, the 
collisions between the particles are less frequent, and the particles 
travel a considerable distance between those impacts. Now, 
although at the instant of impact they send out forced vibrations 
of all sorts of periods, these quickly die away, and thus leave the 
particle free between impacts to send out its own natural vibra- 
tions; therefore the spectrum of incandescent vapors consists 
mainly of vibrations that are characteristic of tJie particles them- 
selves, for each has its own particular natural period and has 
plenty of room in which to vibrate. 

411. X-Eays. Another interesting form of radiation is that 
found in the X-rays. The apparatus used for generating these 
rays is shown in Fig. 236. So much has been written in the 
magazines about these rays, and their applications to surgery 
have brought them so prominently before the public, that we 
need only mention some of their most marked characteristics. 
These are: 1. They appear to originate wherever cathode rays 
fall on matter of any kind. 2. They travel in straight lines, but, 
unlike light, they are not reflected or refracted. Thus they can 
not be deviated by a prism or brought to a focus by a lens. 3. 
Unlike the cathode rays, they are not deflected by a magnet. 
4. The depth to which they penetrate material substances is 
nearly proportional to the densities of the substanc»es. Thus 
they pass readily through wood and animal tissues, which are 
opaque to light. 5. They produce fluorescence in some sub- 
stances, like platinum-barium cyanide and calcium tungstate. 



These substances are therefore used in making the screens 
with which shadows of the bones may be seen. 6. They act on 

Fig. 236. A Modern X-Ray Coil and Tube 

a photographic plate, so that shadow pictures of the bones may 
be made. Fig. 237 is such a picture of a hip joint. 7. When 
they fall on an electrically charged body, the charge quickly 

What are these rays? Since they are not refracted or re- 
flected as light is, they probably do not consist of waves. Further, 
they are not deflected by a magnetic field, and therefore they 
probably are not projected charged particles. In order to gain 
some conception as to what they may be, consider the way in 
which they are generated. A negatively charged particle, travel- 
ing with a great velocity, strikes on some sort of matter. Such 
a traveling electron produces, while in motion, an electric current; 
but when its motion is suddenly stopped the current it is producing 
stops also, and this sudden stopping of the current must give the 
ether a quick jar or impulse. This impulse in the ether might be 
likened to that produced in air when a hammer strikes a block of 
wood. There is a sudden, sharp impulse of sound and we hear a 
click. Such an impulse is not a wave, though it is propagated 
like a wave in that it travels with the velocity of sound in air, 



and also in that it can be heard. Similarly, we are led to the con- 
ception that X-rays may be a series of such sudden impulses in 
ether — an ether phenomenon analogous to the sound phenomena 
produced by a Fourth of July celebration. Such a series of 

Fig. 237. X-Ray Photograph 

impulses would travel with the velocity of light, would not be 
reflected or refracted as light is, and would not be deflected by a 
magnet; i.e., it would possess the properties exhibited by a beam 
of X-rays. 

412. White Light. This conception of a series of impulses 
in ether may help us to conceive of the nature of ordinary white 
light. For we have seen how such light consists of a vast complex 


of waves, each originating at one electron in a complex atom. 
We have also noted how the nature of the complex vibration 
changes with every impact of the atom. Therefore we may 
conceive that white light consists of a vast complex of compara- 
tively short wave trains, each containing a few hundred thousand 
waves. Hence a wave front can not be said to exist in white light, 
since there is no plane or line in which all the particles are in the 
same phase at the same time. This same conclusion holds even 
for monochromatic light. Thus we find that the wave fronts of 
which we have talked are only convenient mathematical fictions, 
which make it possible for us to discuss in a rough sort of way 
the marvelously complex phenomena of light. 

413. Properties of Electrons. Before closing this discussion 
it will be well to recall the facts and theories which we have been 
studying, and to see if we can arrangfe them into a satisfactory 
whole. To do this, let us review the main outlines of the argu- 
ment, and then try to show how the results obtained by it may 
lead to clear conceptions as to the mechanism of these phenomena 
of Nature. 1. We have learned that it has been possible to rec- 
ognize the existence of particles smaller than atoms — about ygVir 
of the size of a hydrogen atom. 2. These particles have been 
found to carry negative electric charges, and to travel with high 
velocity. 3. It has also been possible to conceive how their 
oscillations within atoms may produce the waves which we call 
heat and light. 4. We have seen how these charged particles, 
or electrons, seem to be of the same size and nature, no matter 
from what substance they come. 5. We have learned that X-rays 
are generated when these particles strike on matter. 6. We 
have found that electrons are continually being shot out by radio- 
active substances. 

414. Positively Charged Particles. We may noW ask, Are 

positively electrified particles known? Electrons are always 
charged negatively. The answer to this question may be 
obtained from a further study of radio-activity. It is found that 
radio-active substances emit three kinds of particles. These 


three kinds have been named, from the Greek letters, the alpha, 
the beta, and the gamma particles, and they have different prop- 
erties. The beta particles are found to act in many ways like 
the electrons; i.e., they have a mass about y^Vu ^^ that of the hy- 
drogen atom; they travel with a velocity nearly equal to that of 
light, and they carry negative charges. Further, their ability to 
penetrate into substances depends only on the density of the 
substance, being inversely proportional to it. The alpha particles, 
on the other hand, are much larger, travel with a velocity of only 
tV that of light, and carry positive charges. These particles can 
penetrate into substances much less easily than the beta par- 
ticles. However, on account of their greater mass, they possess 
greater kinetic energy than the beta particles. Their mass is 
found to be about the same as that of an atom of helium; i.e., 
about that of two hydrogen atoms. The nature of the gamma 
particles has not yet been determined. 

415. The Theory of Atomic Structure. The phenomena of 
radio-activity are believed to consist in the spontaneous break- 
ing up of the atoms of the radium or of the other substances. 
And since this disintegration produces both positive and negative 
particles, we have to conceive that the atoms consist of both. 
In fact, we should have to conceive that both exist in atoms in 
order that they remain stable, for a large number of negatively 
charged particles would repel one another and not stay long to- 
gether in one group, unless they were held there by some posi- 
tive attracting force. So we are led to believe that the atom con- 
sists of a positively charged particle about which a number of 
electrons are rotating, like the planets about the sun; i.e., we 
imagine that the atom of matter is constructed on the same general 
plan as the solar system, which may thus be considered as an atom 
of the universe. 

Professor J. J. Thomson, to whom we are particularly in- 
debted for the experimental work on which this new theory of 
matter is based, has shown how such complex particles might be 
formed, and what arrangements of electrons about the central, 
positively charged particle are mechanically possible. He has 


compared his results with the order of the chemical elements 
according to their chemical properties as the chemists have ar- 
ranged them, and finds almost entire agreement. While this fact 
is of great importance and of wonderful interest, it must not be 
taken to be a proof that atoms are really so formed. All that we 
can prove is that it is one possible way. 

Those who are interested in the remarkable and rapid prog- 
ress that has been made in the last ten years by science in thus 
prying into the nature of atoms will find a very good account of 
the theory and its consequences in Whethan, Recent Development 
of Physical Science (London, Murray, 1904). The limitations of 
this, our work, make it possible to give only the barest outlines 
of the subject. 

416. Conclusion. We must not, however, be led to think 
that science has now solved the riddle as to the nature of matter. 
For, even if we have discovered the mechanism of atoms, we have 
only pushed the bounds of ignorance one step further back. Though 
we may be able to say that the atom is not indivisible, but is con- 
structed in such and such a way, we have still to show what elec- 
trons are, what the ether is, what an electric charge is, and whence 
they all come. It must, nevertheless, be clear to every one who 
has read this book carefully, that nature is not a vast chaos of 
chance happenings, but a well ordered and governed whole. 
When we study thoughtfully the phenomena about us, we must 
realize that there are some simple and universal principles which 
are manifest in them all. Therefore, let us leave our study with 
this idea: that the universe in which we live is a marvelously 
organized and governed unit. And when we try to imagine how 
such a unit could have been developed, we are compelled to rec- 
ognize that it could not have come to its present perfection if it 
originated in an unthinkable chaos, and organized itself solely 
by the interaction of blind matter and undirected motion. 


Aberration, spherical. 392; chromatic, 411 

Absolute temperature, 143 

Absorption, of radiant heat, 164, of light, 416: 
color produced by. 416; and radiation, 166; 
spectra, 414 

AbeciBsa, 18 

Accelerated motion, relation of d'.stances to times, 
22; graphical representation, 23; laws of, 27 

Acceleration, defined. 21; negat.ve. 26; determina- 
tion of, 28; of gravity, 31; varies wiUi force, 
33; varies with mass, 35; of gravity, same for 
all bodies, 39. 323; angular. 98; towards 
center, 104 

Accidentals, in musical scale. 346 

Achromatic lens. 410 

Activity, rate of doing work, 50 

Air, weight of, 115; density of, 125; pump, 126; 
atmospheric pressure. 117; critical temper- 
ature of, 150; liquid, 173; columns, vibrating, 

Alcohol, boiling point of, 148; critical temperature, 

Altematmg current transmission, 280 

Aluminum, reduction of. 280 

Ammeter. 268 

Ammonia, critical temperature of. 150; use in 
freezing. 174 

Ampere, theory of magnetism. 235; unit of cur- 
rent. 266 

Amplitude, of waves, 302; and intensity of 
sound, 337 

Analysis, spectrum. 412 

Angle, unit. 98; lens. 386; visual. 391 

Angular; units, 98; size of image, 387 

Anode, 287 

Aperture, angle of, 401 

Archimedes' 8 principle, 126 

Arc lamp, 265; regulator for, 282 

Arm of force, 75; of mass, 106 

Armature, relay. 225; motor 231; dynamo. 253 

Astronomical telescope, 393 

Atom, theory of construction of. 430, 445 

Atmosphere, pressure of, 117; water vapor m, 165; 
of the sun, 415 

Audition, limits of, 350 

Axes, coordinate. 18 
Axis, optical, 384 

Bach, Johann Sebastian, 343 

Back pressure in steam engine, 178 

Balance, equal arm, 89 

Balloons, buoyancy of, 129 

Banju, 334 

Barometer, mercurial, 117 

Battery, voltaic, 283; storage. 291 

Beats, cause discord, 350 

Bell, electric 227; sounds of. 365 

Bessel, gravity with pendulum, 323 

Boiling point, defined, 148; of water. 148; of 

alcohol, 148 
Boyle, Robert, air pump. 125; law of. 131 
Brushes, motor, 231; dynamo, 253 
Bunsen, photometer, 399 
Buoyancy, 129 

C5-G-S system of units, 15 

Calorimeter, 145, 275 

Calorie, gram, 144; mechanical equivalent of, 

Camera, pin-hole, 372; photographic. 391 
Candle, standard, 398 
Carbon dioxide, critical temperature. 150; use in 

freezing, 174 
Carborundum, manufacture of, 280 
Cathode. 287; rays, 436 
Cell, voltaic. 217. 283 
CeUo, 334 

Center of mass, 81; determination of, 83 
Centimeter, defined, 15 
Centrifugal force, 105 
Centripetal force, 105 

Charges, moving electric have magnetic field. 238 
Charies. law of. 140 
Chimes, 365 

Chromatic aberration. 411 
Chromic acid cell, 288 
Circuits, divided, 281; electric, 284; magnetia 

Circular motion. 103 
Clarinet, 336 




Climate, cITcct of water on. 153 

Coefficient of expansion, of Rases, 140; of solids, 
143; of liquids. 143: linear. 143 

Coherer. 428 

Cold storage, 174 

Collecting rings. 231 

Color, and wave length, 370; complex, 415; com- 
plementary, 419 

Columbus, Christopher, magnetic declination, 215 

Commutator, motor, 232; dynamo, 253 

Compass, mariner's, 211; declination of, 215 

Complementary color, 419 

Complex waves, 308; tones, 360; color, 415 

Component motion, 67 

Composite machine, mechanical advantage of, 85 

Composition of motions, 57, 67; of forces, 63 

Compound microscope, 397 

Compressed air, for drills, 112, 122 

Concave, mirror, 383; lens, 395 

Condensers, steam engine, 179; electric, 201 

Condensing pump, 113 

Conduction, of heat, 158; electrical, 192 

Conductors', defined, 192; charge resides on out- 
side of, 200 

Conjugate foci, 388 

Conservation of energy, law of, 44, 237 

Convection, of heat, 158 

Convex lens, 376 

Cooling, by expansion of gas, 172; by evaporation, 

Coordinate axes, 18 

Copper, in voltaic cell, 217, 287 

Comet, 336 

Corti, fibers of, 350 

Coulomb, law of electrostatic force, 200 

Counterpoise on driving wheels, 84 

Crane, traveling, 61 

Crests, of waves, 301 

Critical temperature, 150 

Crookes, Sir William, vacuum tube, 436 

Ctesibius, 114 

Currents, electric, 216; possess magnetic field. 219; 
production by cells, 217; unit, 266; induced 
by a magnet, 245; induced by currents, 248; 
laws of, induced, 250; heating effect, 275 

Dark line spectra, 414 

Davy, Humphrey, arc lamp, 265; electrolysis, 289 

Declination, magnetic, 215 

Density, defined, 41; of water equals 1. 41; de- 
termination of, by Archimedes' 8 principle, 129 
of air, 125; and velocity of waves, 306 

Deviation, minimum, 410 

Dew, 152 

Diar)hragm, telephone 250; lens, 302 

Dielectric, 201; strain in, 203 

Diffuse reflection, 379 

Diffusion, 160 

Direction, jxirccption of, 371 

Discord. 352, 364 

Dispersion, 407 

Displacement and Work, 42; electric, 203; pro- 
portional to force in simple harmonic motion, 

Disruptive discharge, oscillations of, 203; starts 
electric waves. 427 

Distance, unit of, 15 

Distillation, 157 

Distinct vision, limit of, 391 

Dollond, John, achromatic lens, 412 

Dominant triad, 343 

Dufay, law of, 194 

Dyes. 418 

Dynamo, principle of, 251; machine. 252; alter- 
nating current, 255; efiSciency of, 271 

Dyne, unit force, 38 

Ear, 350; perception of complex tones by, 364 
Earth, magnetism of, 212, 215; rotation proved by 

pendulum, 324 
Efficiency, 87; of locomotive, 180; depends on 

absolute temperature, 180; of triple expansion 

engine, 182; of gas engine, 183; of the dynamo, 

Elasticity, of fluids, 118; of air, 330; and velocity 

of waves, 306 
Electric charges, positive and negative, 194; at- 
traction and repulsion of, 194; generated by 
* separating unlike bodies, 196; imit, 201 
Electrification, two kinds, 193 
Electric motor, 234; series, shunt, and compound 

wound, 235 
Electric waves, 426 
Electrolysis, discovery of, 218; Faraday's lawB, 

Electromagnetism, discovery of, 218 
Electromagnets, 221, 283 
Electromotive force, 256 
Electrons, properties of, 435, 444 
Electroplating, 290 
Electroscope, gold leaf, 194 
Electro8tatic8,Dufay's law,194; Coulomb's law 200 
Energy, conservation of, 44, 237, 246; measured by 

work, 44; potential, 46; kinetic, 46; heat, 162; 

magnetic, 237; electric, 270; of voltfuc cell, 284 
Engineering units, 50 
Equilibrant, 67 
Equilibrium, of forces, 64; of paralld forces, 78; 

of body free to rotate, 81 
Erg, unit of work, 43; symbol, 43 



Ether, transmits waves, 163; propagates light, 
426; transmits magnetic and electric action, 


Evaporation, 146, 161 

Expansion, by heat, 143 

Eye, image in, 373; far-sighted, 890; 

390, how focused, 389; color 
Eyepiece, 394 

Falling body, 31 

Faraday. Michael, discovery of induced currents, 
245; ring, 256; electrobm 289 

Field, magnetic, 213 

Field magnets, motor, 231; dynamo, 253 

Fifth, 333 

Fizeau, velocity of light, 424 

Flotation, 128 

Fluids, elasticity of, 118; pressure transmitted by 
119; pressure due to weight of, 121 

Flute, 336 

Fly-wheels, 97; effectiveness of, 103 

Focal length, 377; conjugate, 388 

Focus, 377; principal, 385; conjugate, 388 

Footrpound, 50 

Force, relation to mass and acceleration. 37; unit 
defined. 38; symbol for, 39; vectors, 63; 
centripetal, 105; magnetic lines of, 214; di- 
rection of, 220; and displacement in simple 
harmonic motion, 319 

Force constant, in simple harmonic motion, 321 

Force pump, 114 

Forced vibrations, 349; of electrons, 441 

Foucault, pendulum, 324 

Franklin, Benjamin, theory of electricity, 197; and 
lightnmg, 204 . 

Fringes, interference, 406 

Fundamental, of string, 362 

Galileo, acceleration of gravity same for all bodies 
39; air has weight, 115; thermometer, 138; 
telescope, 397; velocity of light, 423 

Galvani, 217 

Galvanometer, 227; D' Arson val, 228 

Gas engine, 182 

Gases, no free surface, 124; expand indefinitely. 
124; Boyle's law, 131; change of volume at 
constant pressure, 139; change of pressure at 
constant volume, 140; coefficient of expan- 
sion of, 140; diffusion of, 160; pressure of, 161; 
kinetic hypothesis. 162; effect of heating, 162; 
heated when compressed, 172; cooled when 
they expand and do work, 172 

Gay Lussac, law of, 140 

Gilbert, William, De Magnete, 192; electrifica- 
tion, 192; terrestrial magnetiism, 211 

Gram calorie, defined, 144; mechanical equiva- 
lent of, 171 

Gram, unit of inass, 37 

Graph, defined, 18 

Gravity, acceleration of, same for all bodies, 39; 
center of, 81; measured by pendulum, 323 

Gravity cell, 288 

Gray, Stephen, conduction, 192 

Guericke, Otto von, air pump, 125; electric re- 
pulsion, 19? 

Gyration, radius of, 103 

Handd, pure intonation, 347 

Harmonic motion, 317 

Harmonics, of string^, 362 

Harmony, 357 

Heat, quantity of, unit of, 144; specific, 144; 
latent, 152; mechanical equivalent, 171; con- 
duction and convection of, 158; radiant, 160 
162; absorption of, 164; and light, 167; a 
form of kinetic energy, 162; enei^ consumed 
m engine, 179; Ion in electrical transmis- 
sion, 277 

Heating, effect of current, 276; electric. 280; sys- 
tems, hot water. 169; hot air, 169; steam, 169 

Heat waves, emitted by bodies at all tempera- 
tures, 164; more intense and complex at 
high temperatures, 167 

Helmholtz, Heinrich von, musical scale, 345; 
resonator, 363; color vision, 417 

Henry, Joseph, invents telegraph, 222 

Hero of Alexandria, turbine, 185 

Hertz. Heinrich. electric waves. 237, 427 

Homogeneous waves, 308 

Hooke, Robert, air pump, 125 

Horse-power, 51; value in ergs. 51 

Horse-power hour, value in ergs, 180 

Humidity, 151 

Hydraulic machines, 120 

Hydrogen, critical temperature, 150; in the vol' 
taic cell, 285; charge of an atom of, 439 

H\i)othesis, 91 

Ice, manufacture of, 174 

Illumination, intensity of, 398 

Ions, in electrolysis, 286; hypothesis. 2S6; charge 

on, 439 
Image, by pin-hole camera. 371; of a point source. 

385; construction of. 386; size and distance 

of, 387; ^^rtuaI. 388 
Impressed period, 331 
Incandescent lamp, 272 
Incidence, angle of. 378 
Inclination, ang.e of or slope, 19 
Inclined plane, 66 



Index of refraction 37a 

Indicator dia$;rair, 178; gauge for making, 188 

Indicator steam 188 

Induced charge, 199 

Induced currents by magnet, 245; by current, 248 

Induction coil, 255 

Induction, dectrostatic charging by, 199 

Inertia, 47; moment of, 101; determination of 

moment of, 102 
Insulators, electric, 192 . 
Intensity of sound depends 'on amplitude, 337; 

law of inverse squares, of, 399 
interferenre frin«e3 404; wave leng.h measured 

by. 406 

Jack screw, 88 

Jar. Leyden, 201 

Joule, James Prescott, mechanical equivalent of 

heat, 171; heating effects of current. 275; 

law of electric heating, 276 

Kilogram, international standard. 37 
Kilogram-meter, 51; force, 51 
Kinetic energy, equation for, 47; heat a form of,162 
Kirchhoff. Gustav, absorption in spectrum, 414 

Latent heat, of steam, 152; of water, 152 

Laws, definition of, 91 

Leclanche cell, 288 

Lens, crystalline in eye, 373; action of in image 

formation, 373; angle; 386; convex, 376; 

concave, 395 
Lenz, Heinrich, law of, 250 
Lever, 75; mechanical advantage, 76; work done 

by, 76; principle, 77 
Leyden jar, 201 
Lifting magnets, 283 

Light, a wave motion, 370; velocity of, 423 
Light waves, red appear at a temperature of, 520 

C, 167; similar to heat waves, 167; wave 

length of, 406; origm of, 435 
Lightning, 205 
Line loss, 271 
Lines of force, magnetic, 213; current induced 

when number is changed, 247 
Liquefaction of gases, 173 
Liquids in equilibrium, laws of< 124; expansion 

of by heat, 143 
Locfomotive engine, 175; operation of, 176; work 

done by. steam in, 177; efficiency of, 180 
Lodestone. 211 

Longitudinal waves, 307; sound waves arc, 329 
Loops, in stationary waves, 311 

Machines, law of, 86; mechanical advantajjc from, 
87; hydraulic, 120; dynamo-electric. 233, 252 

Magdeburg hemispheres, 125 

Magnetos, 254 

Magnets. 216; lifting. 283 

Magnetic, curves, 213; field. 214; lines of force, 

214; circuit, 216; declination, 215; unit pole, 

216; force, law of, 216; field of moving 

charges, 236; system, energy of, 237 
Magnification, of telescope, 394 
Major scale, 344 
Mass, 36; unit of, 37; rdation to force and accel, 

eration, 37; relation to weight, 40; center of. 

81; moment of, 106; in simple harmonic. 

motion, 321 
Masses, comparison of, by units of the same kind, 

35; by forces that give same acceleration, 

36; by weights. 40 
Maxwell, James Gierke, dectric waves, 237 
Mechanical advantage, of inclined plane, 66; of 

lever, 76; of composite machine, 85; from 

law of machines, 87; of screw, 88 
Mechanical equivalent of heat, 171 ■ 
Mdting. 152 

Mercury, thermometer, 142; spectrum of, 409 
Mersenne, laws of strings, 334 
Meter, international standard, 15 
Microphone, 259 

Microscope, simple, 391; compound, 397 
Minimum deviation, 410 
Mirror, plane, 378; image in, 378; concave, 382 
Molecules, 162 

Moment of force, 75; of mass, 106 
Moment of inertia, 101; determination of, 102 
Momentum, change of measures force, 38 
Morse, Samud F. B.. introduces telegraph, 223; 

alphabet, 223 
Motion, uniform, 19; uniformly accelerated, 21; 

trandatory, 29; rotary, 29, 102; Newton's 

laws of, 47; wave, 302; simple harmonic, 317 
Motions, composition of, 57; resolution of, 62 
Motor dectric. 230; alternating current, 258; 

synchronous, induction, 258 
Musical, tone, 331; intervals, 333; scale, 342; 

tones, complexity of, 360 
Musschenbroek, Leyden jar, 201 

Natural period, 331 

Ne\vton, Sir Isaac, acceleration of gravity, same 
■ for all bodies, 40; laws of motion, 47; Prin- 

cipia, 47; gravity with pendulum, 323; 

color, 403 
Nodes, in stationary waves, 311 
Noise, 331 

Objective, of telescope, 394 
Octave, 333 



Oersted, Hans ChriBtian, discovery of dectro- 
magnetism, 218 

Ohm. Georg Simon, 266; unit of resistance, 266; 
law of, 267 

Opera glass, 396 

Optical axis, 384 

Ordinate, 18 

Organ pipe. 335 

Origin, of coordinates, 18; of waves, 300; of 
light waves, 435 

Oscillatory dfecharge, 203; starts electric waves, 

Overtones, 362; beats among, 365 

Oxygen, critical temperature, 150; and combus- 
tion. 244 

Page, Charles G.. inventor of induction coil. 

Paints. 418 

Parallel, forces, 78; circuits. 272; beam, 384 
Parallelogram of motions, 59 
Pascal, 116; principle of, 119 
Pendulum, proof that gravity acceleration same 

for all bodies, 40; compensated, 157; law of. 

321; uses of. 323; Foucault. 324; equation 

for. 324 
Period of vibration, 305; in simple harmonic 

motion, 320; relation to force constant and 

mass. 321; of pendulum, 323; natural and 

impressed, 331 
Permeability. 214 
Perpetual motion impossible, 43 
Phase, of waves, 304 
Photo-chemical rays, 430 
Photometer, 398 

Piano, 332; overtones of strings. 363 
Pigments, 418 
Pm-hole camera, 372, 382 
Pitch. 332; international standard, 349 
Pisa, leaning tower of, 39 
Polarization, electrostatic, 197 
Pole, magnetic, 211; of earth. 215 
Potential, difference of, 268 
Potential energy, 46 
Pound-force, pound-weight, 50 
Power, rate of doing work. 50; horse. 51; electric, 

268; transmission of. 277 
Pressure, atmospheric, 118; due to weight of 

fluid. 121; and volume m gas, 142; of satu- 
rated vapor, 146; electric, 256 
Prevost, theory of exchanges. 164 
Prism, spectrum formed by, 403; dispersion of, 

Proof plane. 195 
Pullleys, 95 

Pumps, lifting, 113; force, 114; air, 125; theory 

of, 126 
Pythagoras, musical scale. 345 

Radian, unit angle defined, 98 

Radiation, of heat. 160. 162; and absorption. 166 

Radium. 439 

Raitoad curves, 107 

Rainbow, 422 

Rays, characteristic, 385; cathode, 436 

Reaction equals action, 48, 105, 246 

Receiver, telephone. 259; wireless telegraph, 

Reflection, laws of. 378; angle of. 378. 376; 
diffuse. 379; total. 382 

Refraction, 374; index of, 375; angle of, 376 

Related tones, 365 

Relay telegraph. 224 

Repulsion, electric 193 

Resistance, electrical unit of, 266; laws of, 267 

Resistivity, 267; table of, 299 

Resolution, of motions, 62, 67; by microscope, 398 

Resonance, 330; of air columns, 335; of atoms, 415 

Resonators, air columns, 335; Hclmholtz, 363; 
piano strings, 363; atoms, 415 

Resultant motion, 58; force. 64 

Reversible processes. 244 

Rods, vibrating, 334 

Rotation, defined, 29; axis of, 29; and transla- 
tion, units of, compared, 101 

Rowland. Henry A., mechanical equivalent of 
heat, 171; magnetic effect of moving charge, 
236; heating effect of current. 276 

Saturated vapor, pressure and temperature of, 146 

Scientific method. 91 

Screw. 88 

Second, defined. 15 

Self-induction, 263 

Series circuit, 218, 270 

Shape of waves, simple. 302; complex, 309 

Shop, motor and belt driven, 279 

Shunts. 282 

Simple harmonic motion, 307; related to circular 

motion, 317; forces and displacements in, 

318; period, 321 
Sine curve, 307, 319 
Sine, of an angle, 319 
Size, angular of image, 387; linear of image, 387; 

of electrons, 438 
Slope, of a graph, 18; measure of, 19; of a curved 

graph, 25 
Snell, law of refraction, 376 
Sodium, spectrum of. 409 
Solids, expansion of by heat, 143 



Sound, 328; a wave motion, 328; waves longi- 
tudinal, 329; velocity in air, 330; in water, 
340; intensity, 337; quality, 357 

Sounder, telegraph, 224 

Sounding boards, 337 

Spark, electric is oscillatory, 204 

Specific heat, 144 

Spectacles, 390 

Spectroscope, 409 

Spectrum, 407; continuous, 413; bright-line, 409; 
absorption, 414; complete, 429; information, 
' obtained from, 430; dark-line, 414; analysis, 

Spherical aberration, 392 

Spinthariscope, 440 

Stability, 82 

Standards, length, 15; time, 15; mass, 37; pitch 
349; candle. 398 

Standpipe, 123 

Stationary waves, 310 

Steam, 145; pressure and temperature of, 146; 
superheated, 149; latent heat of, 152; engine 
174; work done by, 176; turbine, 184 

Stops, in lens, 392 

Storage battery, 290 

Strings, laws of vibrating, 334; vibrating give 
complex tones, 361 

Subdominant triad, 343 

Submarine boats, 129 

Sun, atmosphere of. 415 

Superheated vapor, 149 

Surface of liquid level, 124 

Symbols, for distance, 15; for time, 15; for veloc- 
ity, 16; for acceleration, 21; for mass, 37; 
for force, 38; for density, 41; for work or 
energy, 43; for angular velocity, 98; for an- 
gular acceleration, 98 

Syphon, 135 

Syren. 333 

Tangent of an an^e, 20 

Telegraph, relay, 224; sounder, 224; key, 226; 

wireless. 428 
Telephone, 259 

Telescope, astronomical, 393; Galileo's 396, 
Temperature, Centigrade scale of, 139; alBolute 

143; critical. 150; depends on kinetic energy 

of molecules 162; efficiency of steam engine 

depends on, 180 
Tempered musical scale, 347 
Thalcs, lodestone, 191 
Theory, 91 
Thermometer. Galileo's, 138; air, 141; mercury, 

Thermostat. 157 

Thomson, J. J., electrons, 439; construction of 

atom, 445 
Three-wire system, 278 
Time, unit of, 15 

Tone quality, and wave shape, 357 
Tones, musical, 331; complex, 360; related, 

Tonic triad. 343 
Tops. 108 
Torricelli. 115 
Torsional vibrations, 327 
Transformer. 256 
Translation, defined, 29; and rotation, units of. 

compared, 101 
Transmitter, telephone, 250 
Transmission, alternating current, 280 
Tcansvene waves. 307 
Triads, tonic, dominant, subdominant, 343; vi- 

bration ratios in, 344 
Trombone, 336 
Troughs, of waves, 301 
Tuning fori:, 335 
Turbines, steam, 184 
Tyndall, John, absorption by water vapor, 165 

Ultra-red, violet, 429 

Uniform motion, equation of, 18 

Unit, length, 15; time, 15; mass, 37; heat, 144; 

electric, charge, 201; magnetic pole, 216 
Units, C-G-S system of, 15 

Vacuum nature abhors, 114; tube, 436 
Vapors, 131; saturated, 146; superheated, 149 
Vector, 58 
Velocity, linear, defined, 16; unit of. 16; angular, 

98; of waves, 305; of sound in air. 330; in 

wat6r, 340; of light, 423; of electric waves, 

Vibrating, strings, laws of, 334; give complex 

tones, 361; flame. 358 
Vibration number. 305 
Vibration, source of waves, 301; time of, 305; 

of pendulum, 321; source of sound, 328; of 

strings and rods. 334; number, 343; forced, 

349; of electrons, 440 
Violin, 334 

Virtual image by lens, 388 
Viision. limit of distinct, 390 
Visual angle, 391 
Volt, 267 

Volta. Alessandro, 217 
Voltaic cell. 217, 283; energy of, 284; pohirisa' 

tion of. 285; local action in, 287; commercial, 

Voltmeter, 268 



Water, density of, 41; boiling point of, 148; 
evaporation of, 145; critical temperature, 
150; latent heat of, 152; and climate, 153; 
vapor, saturated, 148 

Water vapor, formation of, 145; saturated, 148; 
absorption of heat by, 165; effect of in at- 
mosphere on climate, 165 

Watt, unit of electrical power, 269; meters, 270 

V/aves, in water. 163, 300; heat, 163; origin of, 
300; characteristics of, 301; transverse, longi- 
tudmal, 307; stationary, 310; light. 406; 
electric, 426 

Wave front, 384; 444 

Wave length, 302; and phase, 305; and velocity, 
306; and length of strings, 339; of different 
colors, 406; of heat and dectric waves, 

Wave motion, 302; sound is. 328; light is, 370 

Wave shape, and Um» quality, 357 

Weight, 38; relation to mass, 39 

Weight and mass, proportionality proved by fall- 
ing bodies, 39; by pendulum, 323 

Wheatstone bridge, 297 

White light, interference in, 406; nature of, 443 

Windlass, 85 

Wireless telegraphy. 428 

Wiring table, 298 

Work, relation to force and distance, 42; unit of, 
43; measures energy, 44; done by steam in 
an engine, 177; by electric current 268 

X-rays, 440 

Yerkes telescope, 395 

Young. Thomas, color vision, 417 

Zinc^ in voltaic cell, 217. 244, 287; spectrum of, 

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